# Kaplansky Test Problems for $R$-Modules in ZFC

Mohsen Asgharzadeh  Mohsen Asgharzadeh, Hakimiyeh, Tehran, Iran.
<EMAIL_ADDRESS>,  Mohammad Golshani  Mohammad Golshani, School of
Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box:
19395–5746, Tehran, Iran<EMAIL_ADDRESS>and  Saharon Shelah  Saharon
Shelah, Einstein Institute of Mathematics, The Hebrew University of Jerusalem,
Jerusalem, 91904, Israel, and Department of Mathematics, Rutgers University,
New Brunswick, NJ 08854, USA<EMAIL_ADDRESS>
###### Abstract.

Fix a ring $R$ and look at the class of left $R$-modules and naturally we
restrict ourselves to the case of rings such that this class is not too
similar to the case $R$ is a field.

We shall solve Kaplansky test problems, all three of which say that we do not
have decomposition theory, e.g., if the square of one module is isomorphic to
the cube of another it does not follows that they are isomorphic. Our results
are in the ordinary set theory, ZFC. For this we look at bimodules, i.e., the
structures which are simultaneously left $R$-modules and right $S$-modules
with reasonable associativity, over a commutative ring $T$ included in the
centers of $R$ and of $S$.

Eventually we shall choose $S$ to help solve each of the test questions. But
first we analyze what can be the smallest endomorphism ring of an
$(R,S)$-bimodule. We construct such bimodules by using a simple version of the
black box.

###### Key words and phrases:

Black box; bimodules, decomposition theory of modules; endomorphism algebra;
infinitary logic; Kaplansky test problems; set theoretic methods in algebra;
pure semisimple rings, superstable theories.

###### 2010 Mathematics Subject Classification:

Primary: 20K30; 03C60; 16D20; 03C75; Secondary: 03C45; 13L05

The second author’s research has been supported by a grant from IPM (No.
1400030417). The third author’s research has been partially supported by
Israel Science Foundation (ISF) grant no: 1838/19. This is publication 421 of
third author.

###### Contents

  1. 0  Introduction
  2. 1 Outline of the proof of Theorem 0.1
  3. 2 Preliminaries from algebra and logic
  4. 3 Developing the framework of the construction
  5. 4 Any endomorphism is somewhat definable
  6. 5 More specific rings and families ${{\mathfrak{E}}}$
  7. 6 All things together: Kaplansky test problems

##  0\.  Introduction

Throughout this paper ${\bf R}$ is an associative ring with $1=1_{{\bf R}}$,
which is not necessarily commutative nor Noetherian. An ${\bf R}$-module is a
left ${\bf R}$-module, unless otherwise specified. We shall prove here for
example (see Theorem 6.5):

######  Theorem 0.1.

For every ring ${\bf R}$, one of the following conditions hold:

  1. (a)

all ${\bf R}$-modules are direct sum of countably generated indecomposable
${\bf R}$-modules (such rings are called left pure semisimple rings), moreover
such a representation is unique up to isomorphism,

or

  1. (b)

for every cardinal $\lambda>|{\bf R}|$ satisfying
$\lambda=\left(\mu^{\aleph_{0}}\right)^{+}$, and
$0<m(\ast),\ell(\ast)<\omega$, there is an ${\bf R}$-module ${\mathbb{M}}$ of
cardinality $\lambda$ such that

$M\cong M^{n}\ \Leftrightarrow\ n\in\\{m(\ast)k+\ell(\ast):k<\omega\\}.$

We shall prove this theorem in ZFC, the Zermelo–Fraenkel set theory with the
axiom of choice. Theorem 0.1 is new even if we assume Jensen’s combinatorial
principle diamond.

######  Corollary 0.2.

Let $0<m_{1}<m_{2}-1$ and assume that ${\bf R}$ is not pure semisimple. Then
there is an ${\bf R}$-module of cardinality $\lambda$ such that:

${\mathbb{M}}^{n^{1}}\cong{\mathbb{M}}^{n^{2}}\mbox{ iff }\quad m_{1}<n^{1}\
\&\ m_{1}\leq n^{2}\ \&\ [(m_{2}-m_{1})|(n^{1}-n^{2})]$

( $|$ means divides).

This result extends several known statements starting with Corner [2], and a
special case of it, when ${\bf R}$ is the ring of integers, was obtained
recently by Göbel-Herden-Shelah [21, Corollary 9.1(iii)].

Given an algebraic theory, one of the major objectives is to find some
structure theorems for the objects under consideration under various
invariants, say for example up to isomorphism. In 1954, Kaplansky formulated
three test problems (see [28, page 12]), where in his opinion a structure
theory can be satisfactory only if it can solve these problems. He formulated
his questions in the context of Abelian groups as follows:

  1. (I)

If ${\bf G}$ is isomorphic to a direct summand of ${\bf H}$ and ${\bf H}$ is
isomorphic to a direct summand of ${\bf G}$, are ${\bf G}$ and ${\bf H}$
necessarily isomorphic?

  2. (II)

If ${\bf G}\oplus{\bf G}$ and ${\bf H}\oplus{\bf H}$ are isomorphic, are ${\bf
G}$ and ${\bf H}$ isomorphic?

  3. (III)

If ${\bf F}$ is finitely generated and ${\bf F}\oplus{\bf G}$ is isomorphic to
${\bf F}\oplus{\bf H}$, are ${\bf G}$ and ${\bf H}$ isomorphic?

He also noted that these problems can be formulated for very general
mathematical systems, and he also mentioned that problem (I) has an
affirmative answer in set theory, namely the Cantor-Schröder-Bernstein
theorem. It may be worth to note that the first two problems were asked
earlier, in 1948, independently by Sikorski [48] and Tarski [49] in the
context of Boolean algebras, where they answered these problems affirmatively
for countably complete Boolean algebras. Also, Sierpinski asked the cube
problem (see below for its statement) in the context of linear orders, see
[47].

There is a great deal of work on Kaplansky’s test problems over different
algebraic structures. In the sequel we summarize some of them.

For Boolean algebras, the first negative result was proved by Kinoshita [30],
who answered test question (I) in negative for countable Boolean algebras.
Hanf [25] answered question (II) in negative for the same class of structures
and Ketonen [29] solved, among many other interesting things, the Tarski cube
problem by producing a countable Boolean algebra isomorphic to its cube but
not its square. A solution to the Schroeder-Bernstein problem for Banach
spaces is the subject of [24] by Gowers. Very recently, Garrett [18] proved
that every linear order which is isomorphic to its cube, is isomorphic to its
square as well, thus answering Sierpinski’s cube problem for linear orders.

It is known that the problems have positive answer for many classes of Abelian
groups, like finitely generated groups, free groups, divisible groups and so
on, see [17]. Jónsson [26] gave negative answers to test problems (I) and (II)
for countable centerless non-commutative groups and then in [27], he gave a
negative answer to problem (II) for the class of torsion-free Abelian groups
of rank 2. In 1961 Sasiada [35] answered the first problem in negative for the
class of torsion-free groups of rank $2^{\aleph_{0}}$. Corner, in his
breakthrough work [2], proved that any countable torsion-free reduced ring can
be realized as an endomorphism ring of a torsion-free Abelian group and
deduced from this a negative answer to the Kaplansky’s test problems (I) and
(II) for countable torsion-free reduced groups. Later, Corner [3] constructed
a countable torsion-free Abelian group ${\bf G}$, which is isomorphic to ${\bf
G}^{3}$ but not to ${\bf G}^{2}$, thereby giving a negative answer to the cube
problem, which asks if ${\bf G}$ is isomorphic to ${\bf G}^{3}$, does it
follows that ${\bf G}$ is isomorphic to ${\bf G}^{2}$? As another
contribution, Corner proved that for any positive integer $r$ there exists a
countable torsion-free Abelian group ${\bf G}$ such that ${\bf G}^{m}\cong{\bf
G}^{n}$ if and only if $m\equiv_{r}n$. This property of ${\bf G}$ is called
the Corner pathology. In particular, the ring of integers has a pathological
object. For countable separable $p$-groups, Elm’s theorem provides an
affirmative solution to Kaplansky problems (I) and (II), and Crawley [6], in
1965, showed that this does not extend to the uncountable case by giving a
negative answer to these problems for the case of uncountable separable
$p$-groups. Given any $n\geq 2,$ Eklof and Shelah [16] have constructed a
locally free Abelian group $G$ of cardinality $2^{\aleph_{0}}$ such that
$G\oplus\mathbb{Z}^{n}\cong G$. Very recently, Richard [34] has presented more
friendly examples of Abelian groups equipped with the pathological property.
Corner’s ideas were used by a number of mathematicians, to give negative
answers to the first two test questions for some classes of Abelian groups.
For instance, see see [9], [10], [11] [15], [34], and [13].

Shelah’s work [46] deals with the first test problem of Kaplansky in the
category of modules over a general ring, however his results were obtained
under set theoretic assumptions beyond ZFC. In fact, Shelah applied Jensen’s
diamond principle, a kind of prediction principle whose truth is independent
of ZFC, to present rigid-like modules which violate Kaplansky test problem.

Thomé [50] and Eklof-Shelah [15] constructed an $\aleph_{1}$-separable Abelian
group ${\mathbb{M}}$ in ZFC such that the Corner’s ring is algebraically
closed in ${\rm End}({\mathbb{M}})$. Consequently ${\mathbb{M}}$ breaks down
the Kaplansky’s test problem, i.e., it is isomorphic to its cube but not to
its square.

In sum, Kaplansky test problems remained widely open for the category of
modules over general rings.

In this paper, we are interested in the category of modules over a general
ring which is not necessarily commutative nor countable. Working in ZFC, we
answer all three test problems. This continues [46], but can be read
independently. Let us stress that all of our results are in ZFC (without any
extra set theoretic axioms), and that we obtain even stronger results than
those of [46]. For example, see Theorem 0.1. We will do this by using a simple
case of the “Shelah’s black box”, see Lemma 3.43 and Theorem 3.44.

The black boxes were introduced by Shelah in [43] and in a more general form
in [44], where he showed, following some ideas from [38, Ch.VII], that they
follow from ZFC. We can consider black boxes as a general method to generate a
class of diamond-like principles provable in ZFC.

In [36] Shelah proved that any ring ${\bf R}$ satisfies one of the following
two possibilities:

  1. (i)

All modules are direct sums of countably generated modules, or

  2. (ii)

For any cardinal $\lambda>|{\bf R}|$ there is an ${\bf R}$-module of
cardinality $\lambda$ which is not the direct sum of ${\bf R}$-modules of
cardinality $\leq\mu$ for some $\mu<\lambda$.

Shelah’s work in [46] extends (ii) in terms of endomorphism algebras. From
this and in ${\bf V}={\bf L}$, he constructed an ${\bf R}$-module
${\mathbb{M}}$ with prescribed endomorphisms modulo an ideal of small
endomorphisms. Then Shelah found a connection from (ii) to the Kaplansky’s
test problems.

Here, we remove the extra assumption ${\bf V}={\bf L}$ and show that it is
possible to get clause $(b)$ of Theorem 0.1 from the condition (ii).

Here, we lose the $\lambda$-freeness (this is unavoidable, even for Abelian
groups- see Magidor and Shelah [31]). In particular we prove here that for
each $m(\ast)$, there is an ${\bf R}$-module ${\mathbb{M}}$, such that
${\mathbb{M}}\cong{\mathbb{M}}^{n}$ if and only if $n$ divides $m(\ast)$ and
we also answer the other Kaplansky’s test problems which were promised in
[46]. Also here we prove explicitly that the theorems apply to elementary (=
first order) classes of modules which are not superstable.

In the course of the proof of Theorem 0.1, we develop general methods that
allow us to prove the following results in ZFC. These results provide negative
solutions to the Kaplansky’s test problems (I) and (II).

######  Theorem 0.3.

Let ${\bf R}$ be a ring which is not pure semisimple and let $\lambda>|{\bf
R}|$ be a regular cardinal of the form $\left(\mu^{\aleph_{0}}\right)^{+}$.
Then there are ${\bf R}$-modules ${\mathbb{M}}_{1}$ and ${\mathbb{M}}_{2}$ of
cardinality $\lambda$ such that:

  1. i)

${\mathbb{M}}_{1}$, ${\mathbb{M}}_{2}$ are not isomorphic,

  2. ii)

${\mathbb{M}}_{1}$ is isomorphic to a direct summand of ${\mathbb{M}}_{2}$,

  3. iii)

${\mathbb{M}}_{2}$ is isomorphic to a direct summand of ${\mathbb{M}}_{1}$.

######  Theorem 0.4.

Let ${\bf R}$ be a ring which is not pure semisimple and let
$\lambda=\left(\mu^{\aleph_{0}}\right)^{+}>|{\bf R}|$ be a regular cardinal.
Then there are ${\bf R}$-modules ${\mathbb{M}}$, ${\mathbb{M}}_{1}$,
${\mathbb{M}}_{2}$ of cardinality $\lambda$ such that

  1. i)

${\mathbb{M}}\oplus{\mathbb{M}}_{1}\cong{\mathbb{M}}\oplus{\mathbb{M}}_{2}$,

  2. ii)

${\mathbb{M}}_{1}\not\cong{\mathbb{M}}_{2}$.

In particular, theorems 0.3 and 0.4 improve the results from [46], by removing
the use of diamond principle.

As Kaplansky test problems and also our results are related to the
indecomposability of modules, let us also review some background in this
direction. Fuchs [17], with some help of Corner, proved, by induction on
$\lambda$, the existence of an indecomposable Abelian group in many cardinals
$\lambda$ (e.g., up to the first strongly inaccessible cardinal), and even of
a rigid system of $2^{\lambda}$ Abelian groups of power $\lambda$. It was
conjectured at that time that for some “large cardinal” (e.g., supercompact)
this may fail. Corner [4] reduced the number of primes to five and later,
Göbel and May [20] to four, in the following sense. Suppose $R$ is an algebra
over a commutative ring $A$, $\lambda$ is an infinite cardinal, and assume
that $R$, as an $A$-algebra, can be generated by no more than $\lambda$
elements. Let $M=\oplus_{\lambda}R$ and suppose in addition that there is an
embedding $R\hookrightarrow{\rm End}_{A}(M)$ by scalar multiplication. Göbel
and May introduced four $A$-submodules $M_{0},M_{1},M_{2},M_{3}$ of $M$ such
that

$R=\\{\varphi\in{\rm End}_{A}(M):\varphi(M_{i})\subset M_{i}\emph{ for all
}0\leq i\leq 3\\}.$

Shelah [39] then proved the existence in every $\lambda$ (using stationary
sets) of an indecomposable Abelian groups of cardinality $\lambda$ (and even a
rigid system of $2^{\lambda}$ ones). Lately, Göbel and Ziegler [23]
generalized this to ${\bf R}$–modules for “${\bf R}$ with five ideals”.
Answering a question of Pierce, Shelah [40] constructed essentially
decomposable111${\bf G}$ is essentially decomposable if ${\bf G}={\bf
G}_{1}\oplus{\bf G}_{2}$ implies ${\bf G}_{1}$ or ${\bf G}_{2}$ is bounded.
Abelian $p$-groups for any cardinal $\lambda$, which is strong limit of
uncountable cofinality.

Eklof and Mekler [12], using diamond on some non-reflecting stationary set of
ordinals $<\lambda$ of cofinality $\aleph_{0}$, got a $\lambda$-free
indecomposable Abelian group of power $\lambda$. Shelah [42] continued this
where he showed that the diamond can be replaced by weak diamond on a non-
reflecting stationary subset of

$S^{\lambda}_{\aleph_{0}}:=\\{\delta<\lambda:\mathop{\mathrm{cf}}(\delta)=\aleph_{0}\\}$

(so for $\lambda=\aleph_{1}$, $2^{\aleph_{0}}<2^{\aleph_{1}}$ suffices).

Dugas [8] continuing [12] proved, under the assumption ${\bf V}={\bf L}$, the
existence of a strongly $\kappa$-free Abelian group with endomorphism ring
${\mathbb{Z}}$ and then, by using $p$-adic rings, Göbel [19] realized to a
larger family of rings.

Dugas and Göbel [9], continuing [8], [19] and [42], proved that for a Dedekind
domain ${\bf R}$ which is not a field, there are arbitrary large
indecomposable ${\bf R}$-modules, there are arbitrarily large ${\bf
R}$-modules which do not satisfying the Krull-Schmidt cancellation property
and also related their work to the Kaplansky’s test problems, by showing that
${\bf R}$ is not a complete discrete valuation ring if and only if there are
${\bf R}$-modules of arbitrary high rank which do not satisfy Kaplansky’s test
problems. They also showed that every cotorsion-free ring is the endomorphism
ring of some Abelian group. Then in [10], they characterized the rings which
can be represented as ${\rm End}({\mathbb{M}})$ modulo “the small
endomorphism” for some Abelian $p$-group, but as it continues [40] (which
dealt with the case when we want the smallest such ring) the representation of
a ring ${\bf R}$ is by an Abelian group ${\mathbb{M}}$ of a power strong limit
cardinal of cofinality $>|{\bf R}|$. The situation is similar in Dugas and
Göbel [11] where the results of [9] and more are obtained in such cardinals.

Black box enables us to get the results of [9], [10] in more and smaller
cardinals, e.g., $\lambda=\left(|{\bf R}|^{\aleph_{0}}\right)^{+}$. Corner and
Göbel [5] continued this, and using ideas of Shelah have presented a detailed
treatment of construction of Abelian groups (or modules) having preassigned
endomorphism rings, and satisfying additional constraints.

For more information, see the books Eklof-Mekler [14] and Göbel-Trlifaj [22].
The books of Prest [32] and [33] talk about model theory of modules, and in
particular on pure semisimple rings.

##  1\. Outline of the proof of Theorem 0.1

In this section we give an overview of the proof of Theorem 0.1. To this end,
we deal with the problem of choosing a ring ${\bf S}$ (essentially the ring of
endomorphisms we would like), and try to build an ${\bf R}$-module which “has
the endomorphisms for $s\in{\bf S}$ but not many more”.

In Section 3, we develop the setting of our construction and to explain it,
let us first present a simple case. Let ${\bf R}$ be a ring (with unit
$1_{{\bf R}}$) which is not left purely semisimple (i.e., as in Theorem
0.1(b)). Then we consider formulas $\varphi(x)$ of the form

$(\exists
y_{0},\ldots,y_{k_{m}})[\bigwedge\limits_{m<m(\ast)}a_{m}x=\sum\limits_{i<k_{m}}b_{m,i}y_{i}]$

where $\\{a_{m},b_{m,i}\\}$ are elements of the ring ${\bf R}$. For any left
${\bf R}$–module ${\mathbb{M}}$ the set
$\varphi({\mathbb{M}})=\\{x:{\mathbb{M}}\models\varphi(x)\\}$ is not
necessarily a sub-module (as ${\bf R}$ is not necessarily commutative), but it
is an additive subgroup of ${\mathbb{M}}$. Now, our assumption implies that
for some sequence $\bar{\varphi}=\langle\varphi_{n}(x):n<\omega\rangle$, each
$\varphi_{n}$ is as above. For every $n$ and ${\mathbb{M}}$ we have
$\varphi_{n+1}({\mathbb{M}})\subseteq\varphi_{n}({\mathbb{M}})$, and for some
${\mathbb{M}}$, the sequence
$\langle\varphi_{m}({\mathbb{M}}):m<\omega\rangle$ is strictly increasing.
Without loss of generality$~{}k_{n}<k_{n+1}$ and:

$\varphi_{n}(x)=(\exists\;y_{0},\ldots
y_{k_{n-1}-1})(\bigwedge\limits_{m<n}a_{m}x=\sum\limits_{i<k_{m}}b_{m,i}y_{i}).$

Let ${\mathbb{N}}_{n}$ be the ${\bf R}$–module generated by $x^{n},y^{n}_{i}$
($i<k_{m}$) freely, except the following relations

$a_{m}x^{n}=\sum\limits_{i<k_{n}}b_{m,i}y^{n}_{i}\quad\forall m<n.$

Let $g_{n}$ be the homomorphism from ${\mathbb{N}}_{n}$ into
${\mathbb{N}}_{n+1}$ mapping $x^{n}$ to $x^{n+1}$ and $y^{n}_{i}$ to
$y^{n+1}_{i}$ (for $i<k_{n}$). Let

${\mathfrak{e}}=\langle{\mathbb{N}}_{n},x_{n},g_{n}:n<\omega\rangle=\langle{\mathbb{N}}^{\mathfrak{e}}_{n},x^{{\mathfrak{e}}}_{n},g^{\mathfrak{e}}_{n}:n<\omega\rangle.$

Next, we define the following subgroups of ${\mathbb{M}}$:

$\varphi_{n}({\mathbb{M}})=\varphi^{\mathfrak{e}}_{n}({\mathbb{M}}):=\\{f(x):f\mbox{
is a homomorphism from }{\mathbb{N}}_{n}\mbox{ to }{\mathbb{M}}\\}.$

In Definition 3.33 we will define the notion of “semi-nice construction”,
which in particular gives a sequence $\bar{{\mathbb{M}}}$ of ${\bf
R}$-modules, and in Theorem 3.44 we show that the existence of a semi-nice
construction follows from a suitable version of black boxs. It gives us an
${\bf R}$–module ${\mathbb{M}}=\bigcup\limits_{\alpha}{\mathbb{M}}_{\alpha}$
such that every endomorphism ${{\mathbf{f}}}$ of ${\mathbb{M}}$ is in a
suitable sense trivial: on general grounds, ${{\mathbf{f}}}$ maps
$\varphi_{n}({\mathbb{M}})$ into itself, hence it maps
$\varphi^{\mathfrak{e}}_{\omega}({\mathbb{M}})=:\bigcap\limits_{n<\omega}\varphi_{n}({\mathbb{M}})$
into itself. So it induces an additive endomorphism
$\hat{{\mathbf{f}}}_{n}={{\mathbf{f}}}\restriction(\varphi_{n}({\mathbb{M}})/\varphi^{\mathfrak{e}}_{\omega}({\mathbb{M}}))$
of the Abelian group
$\varphi_{n}({\mathbb{M}})/\varphi^{\mathfrak{e}}_{\omega}({\mathbb{M}})$. By
the construction, for some $n=n({\mathbf{f}})$, the mapping
$\hat{{\mathbf{f}}}_{n}$ is of the “unavoidable” kind: multiplication by some
$a$ from the center of ${\bf R}$.

The point is that for undesirable ${{\mathbf{f}}}$, we will be able to find
$x_{n}$ and $x$ such that $x-x_{n}\in\varphi_{n}({\mathbb{M}})$ but for no
$y\in{\mathbb{M}}$ do we have:

$n<\omega\quad\Rightarrow\quad
y-{{\mathbf{f}}}(x_{n})\in\varphi_{n}({\mathbb{M}}).$

So, omitting countable types helps. But, we would like to have more: not to
generalize “the Abelian group ${\mathbb{M}}$ has no endomorphism except
multiplication by $a\in{\mathbb{Z}}$”, but to generalize “the Abelian group
${\mathbb{M}}$ has a prescribed endomorphism ring ${\bf S}$”. For this, we
consider a fixed pair $({\bf R},{\bf S})$ of rings and a commutative subring
${\bf T}$ of the center of ${\bf R}$ and the center of ${\bf S}$, and work
with $({\bf R},{\bf S})$-bimodules, where by an $({\bf R},{\bf S})$-bimodule,
we mean a left ${\bf R}$-module ${\mathbb{M}}$ which is a right ${\bf
S}$-module and satisfies the following equations: $\forall r\in{\bf R}$,
$\forall s\in{\bf S}$, $(rx)s=r(xs)$ and $\forall t\in{\bf T}$, $tx=xt$. We
would like to build a bimodule ${\mathbb{M}}$ such that all of its ${\bf
R}$–endomorphisms (i.e., endomorphisms as an ${\bf R}$–module) are, in a
sense, equal to left multiplication by a member of ${\bf S}$. To be able to
construct such an ${\mathbb{M}}$, we need to have
${\mathfrak{e}}=\langle{\mathbb{N}}^{\mathfrak{e}}_{n},x^{\mathfrak{e}}_{n},g^{\mathfrak{e}}_{n}:n<\omega\rangle$,
but now ${\mathbb{N}}^{\mathfrak{e}}_{n}$ is a bimodule,
$x^{\mathfrak{e}}_{n}\in{\mathbb{N}}^{\mathfrak{e}}_{n}$ and
$g^{\mathfrak{e}}_{n}$ is a bimodule homomorphism from
${\mathbb{N}}^{\mathfrak{e}}_{n}$ to ${\mathbb{N}}^{\mathfrak{e}}_{n+1}$
mapping $x^{\mathfrak{e}}_{n}$ to $x^{\mathfrak{e}}_{n+1}$. As a first
approximation, let $\varphi^{\mathfrak{e}}_{n}({\mathbb{M}})$ be

$\\{x:\mbox{ there is an ${\bf R}$-homomorphism from
}{\mathbb{N}}^{\mathfrak{e}}_{n}\mbox{ to }{\mathbb{M}}\mbox{ mapping
}x^{\mathfrak{e}}_{n}\mbox{ to }x\\}.$

Of course, $\varphi^{\mathfrak{e}}_{n}({\mathbb{M}})$ is an additive subgroup
of ${\mathbb{M}}$. We define $\psi^{\mathfrak{e}}_{n}({\mathbb{M}})$ similarly
but using bimodule homomorphism. In fact, we consider a set ${\mathfrak{E}}$
of such ${\mathfrak{e}}$’s.

However, ${\mathbb{N}}^{\mathfrak{e}}_{n}$ is not necessarily finitely
generated as an ${\bf R}$-module. So let us restrict ourselves to bimodules
such that, locally they look like direct sum of ${\bf R}$-modules from some
class ${\mathcal{K}}$. This property is denoted by
$0\leq_{\aleph_{0}}{\mathbb{M}}$ (essentially it can be represented as being
${\mathbb{L}}_{\infty,\aleph_{0}}$-equivalent to such a sum; more fully
$0\leq^{{\rm ads}}_{{\mathcal{K}},\aleph_{0}}{\mathbb{M}}$, see Definition
3.15). Now, if $0\leq_{\aleph_{0}}{\mathbb{M}}$, we let

$\begin{array}[]{r}\varphi^{\mathfrak{e}}_{n}({\mathbb{M}}):=\\{x\in{\mathbb{M}}:\mbox{
there is an ${\bf R}$--homomorphism from ${\mathbb{N}}^{\mathfrak{e}}_{n}$
into ${\mathbb{N}}$ }\\\ \mbox{mapping $x^{\mathfrak{e}}_{n}$ to
$x$}\\},\end{array}$

and

$\begin{array}[]{r}\psi^{\mathfrak{e}}_{n}({\mathbb{M}}):=\\{x:\mbox{ there is
a bimodule homomorphism from ${\mathbb{N}}^{\mathfrak{e}}_{n}$ into
${\mathbb{N}}$ }\\\ \mbox{mapping $x^{\mathfrak{e}}_{n}$ to
$x$}\\}.\end{array}$

Also our complicated set ${\mathfrak{e}}$ makes us to define the set
${\mathbb{L}}^{\mathfrak{e}}_{n}$ of elements of
${\mathbb{N}}^{\mathfrak{e}}_{n}$ whose image under bimodule homomorphism is
determined by the image of $x^{\mathfrak{e}}_{n}$.

In addition, we would like to include in our framework the class of ${\bf
R}$-modules of a fix first order complete theory $T$; this is fine for $T$ not
superstable, but we need to replace the requirement
$0\leq_{\aleph_{0}}{\mathbb{M}}$ by
“${\mathbb{M}}^{\ast}\leq_{\aleph_{0}}{\mathbb{M}}$” and choose
${\mathcal{K}}$ and ${\mathbb{M}}^{\ast}$ appropriate for $T$. For example,
${\mathbb{M}}^{\ast}$ can be any $\aleph_{1}$-saturated model of $T$ and

${\mathcal{K}}:=\\{{\mathbb{N}}:{\mathbb{N}}\mbox{ is an ${\bf R}$--module
such that }{\mathbb{M}}^{\ast}\prec_{{\mathcal{L}}(\tau_{\bf
R})}{\mathbb{M}}^{\ast}\oplus{\mathbb{N}}\\},$

where ${\mathcal{L}}(\tau_{\bf R})$ is the language of ${\bf R}$-modules (see
Definition 2.5). Also, ${\mathbb{N}}$ is not too large, e.g. it has size at
most $\|{\bf R}\|+\|{\bf S}\|.$

Let $\bar{\mathbb{M}}=\langle{\mathbb{M}}_{\alpha}:\alpha\leq\kappa\rangle$ be
a semi-nice construction and let ${\mathbb{M}}_{\kappa}$ be the module defined
by $\bar{\mathbb{M}}$. For any ${\bf
f}:{\mathbb{M}}_{\kappa}\to{\mathbb{M}}_{\kappa}$, $\alpha<\kappa$ and
$n<\omega$, we consider the following principle: For any bimodule homomorphism
${\bf h}$ from ${\mathbb{N}}^{\mathfrak{e}}_{n}$ into ${\mathbb{M}}_{\kappa}$,
and for every $\ell<\omega$ we have

${\bf f}({\bf
h}(x_{n}^{\mathfrak{e}}))\in{\mathbb{M}}_{\alpha}+\varphi^{\mathfrak{e}}_{\ell}({\mathbb{M}}_{\kappa})+\mathop{\mathrm{Rang}}({\bf
h}).$

We refer to this property by saying that the statement
$({\mathord{\mathrm{Pr}}})^{n}_{\alpha}{[}{\mathbf{f}},{\mathfrak{e}}{]}$
holds. In Lemma 4.2 we prove that every ${\bf R}$-endomorphism
${{\mathbf{f}}}$ of the module ${\mathbb{M}}_{\kappa}$ constructed in Section
3 is “somewhat definable” and specifically satisfies
$({\mathord{\mathrm{Pr}}})^{n(\ast)}_{\alpha}[{{\mathbf{f}}},{\mathfrak{e}}]$
(for some $\alpha<\kappa$, $n(\ast)<\omega$ and for all the $\mathfrak{e}$’s
we have taken care of). We show that
$({\mathord{\mathrm{Pr}}})^{n(\ast)}_{\alpha}[{{\mathbf{f}}},{\mathfrak{e}}]$
implies a stronger version, denoted by
$({\mathord{\mathrm{Pr}}}1)^{n(\ast)}_{\alpha,z}[{{\mathbf{f}}},{\mathfrak{e}}]$
where $z\in{\mathbb{L}}^{\mathfrak{e}}_{n}$. More explicitly, if $h$ is a
bimodule homomorphism from ${\mathbb{N}}^{\mathfrak{e}}_{n}$ into
${\mathbb{M}}_{\kappa}$,
$({\mathord{\mathrm{Pr}}}1)^{n(\ast)}_{\alpha,z}[{{\mathbf{f}}},{\mathfrak{e}}]$
implies that

${\bf
f}(h(x^{\mathfrak{e}}_{n}))-h(z)\in{\mathbb{M}}_{\alpha}+\varphi^{\mathfrak{e}}_{\omega}({\mathbb{M}}_{\kappa}).$

This is the subject of Lemma 4.7.

In Section 5 we try to say more: in ${\mathbb{M}}_{\kappa}$ every endomorphism
is in some suitable sense equal to one in a ring ${{dE}}$, whose definition is
given in Lemma 5.24(iii). Here, we remark that the “in some sense equal”
means: for each $n$ we restrict ${{\mathbf{f}}}$ to an additive sub-group
$\varphi^{\mathfrak{e}}_{n}({\mathbb{M}}_{\kappa})$ (closed under
${{\mathbf{f}}}$), divide by another
$\varphi^{\mathfrak{e}}_{\omega}({\mathbb{M}}_{\kappa})$ and take direct
limit; on the top of this we have an “error term”: we have to divide by a
“small” sub-module of ${\mathbb{M}}_{\kappa}$, which means of cardinality
$<\lambda=\|{\mathbb{M}}_{\kappa}\|$. A better result is: we divide the ring
of such endomorphisms by the ideal of those with “small” range and then even
“compact ones” which are essentially of cardinality
$\leq\mathop{\mathrm{sup}}\\{\|{\mathbb{M}}\|:{\mathbb{M}}\in{\mathcal{K}}\\}$.
We close Section 5 by proving the following result.

######  Corollary 1.1.

Suppose ${\bf S}$ is a ring extending $\mathbb{Z}$ such that $({\bf S},+)$ is
free and let ${\bf R}$ be a ring which is not pure semisimple. Let ${\bf D}$
be a field such that $p:=\operatorname{char}({\bf D})|\operatorname{char}({\bf
R})$ and set ${\mathbb{Z}}_{p}:={\mathbb{Z}}/p{\mathbb{Z}}$ if $p>0$ and
${\mathbb{Z}}_{p}:={\mathbb{Z}}$ otherwise. Let $\Sigma$ be a set of equations
over ${\bf S}$ not solvable in ${{\bf
D}}\mathop{{\otimes}}_{{\mathbb{Z}}_{p}}({\bf S}/p{\bf S})$. Then for
${\mathbb{M}}$ which is strongly nicely constructed, $\Sigma$ is not solvable
in ${\rm End}({\mathbb{M}})$.

Let us now explain how the above constructions can be applied to prove Theorem
0.1 (for more details see Section 6). We first introduce a ring ${\bf S}_{0}$,
it is incredibly easy compared to ${\bf S}$. To this end, let ${\bf T}$ be the
subring of ${\bf R}$ which 1 generates. Let ${\bf S}_{0}$ be the ring
extending ${\bf T}$ generated by
$\\{{\mathcal{X}}_{0},\ldots,{\mathcal{X}}_{m(\ast)-1},{\mathcal{W}},{\mathcal{Z}}\\}$
freely except:

$(\ast)_{1}$:

${\mathcal{X}}^{2}_{\ell}={\mathcal{X}}_{\ell}$,
${\mathcal{X}}_{\ell}{\mathcal{X}}_{m}=0$ ($\ell\neq m$),
$1={\mathcal{X}}_{0}+\ldots+{\mathcal{X}}_{m(\ast)-1}$,
${\mathcal{X}}_{\ell}{\mathcal{W}}{\mathcal{X}}_{m}=0$ for $\ell+1\neq m\
\mod\ m(\ast)$,
${\mathcal{W}}^{m(\ast)}=1$,
${\mathcal{Z}}^{2}=1$,
${\mathcal{X}}_{0}{\mathcal{Z}}(1-X_{0})={\mathcal{X}}_{0}{\mathcal{Z}}$,
$(1-{\mathcal{X}}_{0}){\mathcal{Z}}{\mathcal{X}}_{0}=(1-{\mathcal{X}}_{0}){\mathcal{Z}}$.

For an integer $m$ let $[-\infty,m):=\\{n:n\mbox{ is an integer and
}~{}n<m\\}$. To each $\eta\in{}^{[-\infty,m)}\omega$ we assign
$\eta\restriction
k:=\eta\restriction[-\infty,\mathop{\mathrm{min}}\\{m,k\\})$. We look at

$\begin{array}[]{lll}W_{0}:=&\big{\\{}\eta:&\eta\mbox{ is a function with
domain of the form }[-\infty,n)\\\ &&\mbox{and range a subset of
}\\{1,\ldots,m(\ast)-1\\}\mbox{ such that}\\\ &&\mbox{for every small enough
}m\in{\mathbb{Z}},\ \eta(m)=1\big{\\}}\end{array}$

and $W_{1}:=W_{0}\times\\{0,\ldots,m(\ast)-1\\}$. Let $\bf D$ be a field such
that

${\bf T}\mbox{ is finite }\quad\Rightarrow\quad\operatorname{char}({\bf
D})\mbox{ divides }|{\bf T}|.$

So, ${\bf D}\otimes{\bf S}$ is the ring extending $\bf D$ by adding ${\bf
D},{\mathcal{X}}_{0},\ldots,{\mathcal{X}}_{m(\ast)-1},{\mathcal{W}},{\mathcal{Z}}$
as non commuting variables freely except satisfying the equations in
$(\ast)_{1}$. Let ${\mathbb{M}}^{\ast}={}_{\bf D}{\mathbb{M}}^{\ast}$ be the
left ${\bf D}$-module freely generated by $\\{x_{\eta,\ell}:\eta\in W_{0},\
\ell<m(\ast)\\}.$ We make ${}_{\bf D}{\mathbb{M}}^{\ast}$ to a right $({\bf
D}\mathop{{\otimes}}{\bf S}_{0})$-module by defining $xz$ for $x\in{}_{{\bf
D}}{\mathbb{M}}^{*}$ and $z\in{\bf S}_{0}$, so it is enough to deal with
$z\in\\{{\mathcal{X}}_{m}:m<m(*)\\}\cup\\{{\mathcal{Z}},{\mathcal{W}}\\}$. Let
$x=\sum\limits_{\eta,\ell}a_{\eta,\ell}x_{\eta,\ell}$ where $(\eta,\ell)$ vary
on $W_{0}$ and $a_{\eta,\ell}\in{\bf D}$ and $\\{(\eta,\ell):a_{\eta,\ell}\neq
0\\}$ is finite and we let
$\sum\limits_{\eta,\ell}a_{\eta,\ell}x_{\eta,\ell})z=\sum\limits_{\eta,\ell}a_{\eta,\ell}(x_{\eta,\ell}z),$
where

$\begin{array}[]{rcl}x_{\eta,\ell}{\mathcal{X}}_{m}&:=&\left\\{\begin{array}[]{ll}x_{\eta,\ell}&\mbox{if
}\ell=m,\\\ 0&\mbox{if }\ell\neq m,\end{array}\right.\\\ \\\
x_{\eta,\ell}{\mathcal{Z}}&:=&\left\\{\begin{array}[]{ll}x_{\eta{}^{\frown}\\!\langle\ell\rangle,0}&\mbox{if
}\ell>0,\\\ x_{\eta\restriction[-\infty,n-1),\eta(n-1)}&\mbox{if
}\ell=0,\mbox{ and
}(-\infty,n)=\mathop{\mathrm{Dom}}(\eta),\end{array}\right.\\\ and\\\
x_{\eta,\ell}{\mathcal{W}}&:=&x_{\eta,m}\ \mbox{ when }m=\ell+1\ \mod\
m(\ast).\end{array}$

We get a $({\bf D},{\bf S})$-bimodule. Let ${}_{\bf
D}{\mathbb{M}}^{\ast}_{\ell}=\\{\sum\limits_{\eta}d_{\eta,\ell}x_{\eta,\ell}:\
\eta\in W_{0}$ and $d_{\eta,\ell}\in{\bf D}\\}$, so ${}_{\bf
D}{\mathbb{M}}^{\ast}=\mathop{{\bigoplus}}\limits_{\ell=0}^{m(\ast)-1}{}_{\bf
D}{\mathbb{M}}^{\ast}_{\ell}$.

Now we are ready to define ${\bf S}$, but in addition $\sigma=0$ when
${\mathbb{M}}^{\ast}_{{\bf D}}\sigma=0$ for every ${\bf D}$ and every $({\bf
D},{\bf S}_{0})$-bimodule ${}_{\bf D}{\mathbb{M}}^{*}$ as defined above.

We will prove that ${\bf S}$ is a free ${\bf T}$-module. This allow us to
apply Corollary 1.1. Look at ${\mathbb{P}}$, the bimodule induced by the semi-
nice construction equip with the strong property presented in above. Let
${\mathbb{P}}_{\ell}={\mathbb{P}}{\mathcal{X}}_{\ell}$. It follows easily that
$\mathop{{\bigoplus}}\limits^{m(\ast)-1}_{\ell=1}{}_{\bf
R}{\mathbb{P}}_{\ell}\cong({}_{\bf R}{\mathbb{P}}_{0})^{m(\ast)-1}.$ It is
enough to show ${}_{\bf R}{\mathbb{P}}^{k}_{0}\not\cong{}_{{\bf
R}}{\mathbb{P}}_{0}$ for all $1<k<m(\ast)-1$. Assume $k$ is a counterexample.
We apply Corollary 1.1 to find a field ${\bf D}$ and ${\mathcal{Y}}\in{\bf
D}\mathop{{\otimes}}\limits_{\bf T}{\bf S}$ satisfying the following
equations:

$(*)_{2}$ :

${\mathcal{Y}}\restriction{}_{\bf D}{\mathbb{M}}^{\ast}_{0}$ is an isomorphism
from ${}_{\bf D}{\mathbb{M}}^{\ast}_{0}$ onto
$\mathop{{\bigoplus}}\limits_{\ell=1}^{k}{}_{\bf
D}{\mathbb{M}}^{\ast}_{\ell}$,

${\mathcal{Y}}\restriction\mathop{{\bigoplus}}\limits_{\ell=1}^{k}{}_{\bf
D}{\mathbb{M}}^{\ast}_{\ell}$ is an isomorphism from
$\mathop{{\bigoplus}}\limits_{\ell=1}^{k}{}_{\bf D}{\mathbb{M}}^{\ast}_{\ell}$
onto ${}_{\bf D}{\mathbb{M}}^{\ast}_{0}$,

${\mathcal{Y}}\restriction\mathop{{\bigoplus}}\limits^{m(\ast)}_{\ell=k+1}{}_{\bf
D}{\mathbb{M}}^{\ast}_{\ell}$ is the identity

${\mathcal{Y}}^{2}=1$.

Then, we look at the following sets:

$\begin{array}[]{ll}w_{\eta,\ell}&:=\\{(\nu,m)|(\nu,m)=(\eta,\ell)\mbox{ or
}\eta^{\frown}\langle\ell\rangle\trianglelefteq\nu~{}\mbox{and}~{}m<m(*)\\}\\\
u_{\ell}&:=\\{(\eta,m)|m=\ell,\ \eta\in w_{0}\\}\\\ w^{m}_{\eta,\ell}&:=\
w_{\eta,\ell}\cap u_{m}\\\
w^{[1,n]}_{\eta,\ell}&:=w_{\eta,\ell}\cap\bigcup\limits_{m\in[1,n]}u_{m}.\end{array}$

For any $u\subseteq\\{(\eta,\ell):\eta\in W_{0},\ \ell<m(\ast)\\}$, we let
${\mathbb{N}}_{u}$ be the sub ${\bf D}$-module of ${}_{\bf
D}{\mathbb{M}}^{\ast}$ generated by $\\{x_{\eta,\ell}:(\eta,\ell)\in u\\}$.
For every large enough finite subset $v\subseteq w_{\eta,\ell}$, we show the
following is well defined:

${\bf
n}_{\eta,\ell}:=\dim\left(\frac{{\mathbb{N}}_{w^{0}_{\eta,\ell}}}{{\mathbb{N}}_{w^{0}_{\eta,\ell}\setminus
v}}\right)-\dim\left(\frac{{\mathbb{N}}_{w^{[1,k]}_{\eta,\ell}}}{{\mathbb{N}}_{w^{0}_{\eta,\ell}\setminus
v}{\mathcal{Y}}}\right).$

These integers are independent from the first factor: Suppose $\eta$, $\nu\in
w_{0}$ and $\ell\in\\{0,1,\ldots,m(\ast)-1\\}$ then ${\bf n}_{\eta,\ell}={\bf
n}_{\nu,\ell}$ so we shall write ${\bf n}_{\ell}$. These numbers are such that
they satisfy the following equations:

${\bf n}_{\ell}=\left\\{\begin{array}[]{ll}0&\mbox{if }\ell\in[1,k]\\\ {\bf
n}_{0}+{\bf n}_{1}+\ldots+{\bf n}_{m(\ast)-1}&\mbox{if
}\ell\in[k+1,m(\ast))\emph{ or }\ell=0\end{array}\right.$

Finally, we use this equation to derive the following contradiction

$\sum^{m(\ast)-1}_{\ell=1}{\bf n}_{\ell}=\frac{k}{m(\ast)-1}.$

##  2\. Preliminaries from algebra and logic

In this section we provide some preliminaries from algebra and logic that are
needed for the rest of the paper. Let us start by fixing some notations that
we will use through the paper. Recall that ${\bf R}$ is a ring, not necessary
commutative, with $1=1_{\bf R}$.

######  Convention 2.1.

By $\mathop{\mathrm{Cent}}(-)$ we mean the center of a ring $(-)$. The rings
${\bf S}$ and ${\bf R}$ are with $1$, and we always assume that ${\bf
T}:=\mathop{\mathrm{Cent}}({\bf R})\cap\mathop{\mathrm{Cent}}({\bf S})$ is a
commutative ring.222Indeed, in our applications we have ${\bf T}={\bf
R}\cap{\bf S}$ (see Section 6, below).

######  Definition 2.2.

An $({\bf R},{\bf S})$-bimodule ${\mathbb{M}}$ is a left ${\bf R}$-module and
right ${\bf S}$-module such that for all $r\in{\bf R},x\in{\mathbb{M}}$ and
$~{}s\in{\bf S}$ we have $(rx)s=r(xs)$ and that $tx=xt$ for all $t\in{\bf T}$.
When the pair $({\bf R},{\bf S})$ is clear from the context, we refer to
${\mathbb{M}}$ as a bimodule.

######  Convention 2.3.

We use ${\mathbb{K}}$, ${\mathbb{M}}$, ${\mathbb{N}}$ and ${\mathbb{P}}$ to
denote bimodules (or left ${\bf R}$-modules).

######  Definition 2.4.

Let ${\bf f}:{\mathbb{M}}\rightarrow{\mathbb{N}}$ be a bimodule homomorphism.

  1. (1)

The kernel, $\mathop{\mathrm{Ker}}({\bf f}):=\\{x\in{\mathbb{M}}:{\bf
f}(x)=0\\}$ is a sub-bimodule of ${\mathbb{M}}$.

  2. (2)

The image, $\mathop{\mathrm{Rang}}({\bf f}):=\\{{\bf
f}(x):x\in{\mathbb{M}}\\}$ is a sub-bimodule of ${\mathbb{N}}$.

  3. (3)

If ${\mathbb{N}}$ is a sub-bimodule of ${\mathbb{M}}$ then
${\mathbb{M}}/{\mathbb{N}}:=\\{x+{\mathbb{N}}:x\in{\mathbb{M}}\\}$ is a
homomorphic image of ${\mathbb{M}}$. The mapping $x\mapsto x+{\mathbb{N}}$ is
a homomorphism with kernel ${\mathbb{N}}$.

  4. (4)

For a bimodule ${\mathbb{M}},{\rm End}_{\bf R}({\mathbb{M}})$ is the
endomorphism ring of ${\mathbb{M}}$ as a left ${\bf R}$-module.

In this paper we also consider modules and bimodules as logical structures, so
let us fix a language for them. We only sketch the basic definitions and
results which are needed here, and refer to [1] and [7] for further
information.

######  Definition 2.5.

  1. (1)

An ${\bf R}$-module ${\mathbb{M}}$ is considered as a $\tau_{\bf R}$-structure
where $\tau_{\bf R}=\\{0,+,-\\}\cup\\{{}_{r}H:r\in{\bf R}\\},$ where the
universe is the set of elements of ${\mathbb{M}}$, and $0,+,-$ are interpreted
naturally and ${}_{r}H$ is interpreted as (left) multiplication by $r$ (i.e.,
${}_{r}H(x):=rx$). Then

    1. (a)

Terms of $\tau_{\bf R}$ are expressions of the form $\sum_{i<m}r_{i}x_{i}$,
where $m<\omega,$ $r_{i}$ is in ${\bf R}$ and $x_{i}$ is a variable.

    2. (b)

Atomic formulas of $\tau_{\bf R}$ are equations of the form $t_{1}=t_{2}$,
where $t_{1},t_{2}$ are terms.

  2. (2)

An $({\bf R},{\bf S})$-bimodule is similarly interpreted as a $\tau_{({\bf
R},{\bf S})}$\- structure where $\tau_{({\bf R},{\bf
S})}=\\{0,+,-\\}\cup\\{{}_{r}H:r\in{\bf R}\\}\cup\\{H_{s}:s\in{\bf S}\\}$ and
${}_{r}H,H_{s}$ for $r\in{\bf R},s\in{\bf S}$ are unary function symbol, which
will be interpreted as follows: ${}_{r}H$ as left multiplication by $r$ (i.e.,
${}_{r}H(x):=rx$) and $H_{s}$ as right multiplication by $s$ (i.e.,
$H_{s}(x)=xs$). Then

    1. (a)

Terms of $\tau_{({\bf R},{\bf S})}$ are expressions of the form

$\sum_{i<d}\sum_{h<h_{d}}r_{i,h}x_{i}s_{i,h}$

where $d,h_{d}<\omega,$ $r_{i,h}\in{\bf R}$ $s_{i,h}\in{\bf S}$ and $x_{i}$ is
a variable.

    2. (b)

Atomic formulas of $\tau_{({\bf R},{\bf S})}$ are equations of the form
$t_{1}=t_{2}$, where $t_{1},t_{2}$ are terms.

The next lemma is trivial.

######  Lemma 2.6.

The class of $({\bf R},{\bf S})$-bimodules is a variety, i.e., there are
equations in the language of $\tau_{({\bf R},{\bf S})}$ such that a
$\tau_{({\bf R},{\bf S})}$-structure is an $({\bf R},{\bf S})$-bimodule iff it
satisfies these equations.

Let us now introduce the infinitary languages for modules and bimodules. They
will play essential roles in the sequel.

######  Definition 2.7.

Suppose $\kappa$ and $\mu$ are infinite cardinals, which we allow to be
$\infty$, and let $\tau$ be one of $\tau_{{\bf R}}$ or $\tau_{({\bf R},{\bf
S})}$. The infinitary language $\mathcal{L}_{\mu,\kappa}(\tau)$ is defined so
as its vocabulary is the same as $\tau,$ it has the same terms and atomic
formulas as in $\tau,$ but we also allow conjunction and disjunction of length
less than $\mu$ (i.e., if $\phi_{j},$ for $j<\beta<\mu$ are formulas, then so
are $\bigvee_{j<\beta}\phi_{j}$ and $\bigwedge_{j<\beta}\phi_{j}$) and
quantification over less than $\kappa$ many variables (i.e., if
$\phi=\phi((v_{i})_{i<\alpha})$, where $\alpha<\kappa$, is a formula, then so
are $\forall_{i<\alpha}v_{i}\phi$ and $\exists_{i<\alpha}v_{i}\phi$).

Note that $\mathcal{L}_{\omega,\omega}(\tau)$ is just the first order logic
with vocabulary $\tau.$ Given $\kappa$, $\mu$ and $\tau$ as above, we are
sometimes interested in some special formulas from
$\mathcal{L}_{\mu,\kappa}(\tau)$.

######  Definition 2.8.

  1. (1)

$\mathcal{L}^{\text{cpe}}_{\mu,\kappa}(\tau)$, the class of conjunctive
positive existential formulas, consists of those formulas of
$\mathcal{L}_{\mu,\kappa}(\tau)$ which in their formulation only
$\wedge,\bigwedge_{j<\beta},\exists x$ and $\exists_{i<\alpha}v_{i}$ are used
(where $\beta<\mu$ and $\alpha<\kappa$).

  2. (2)

$\mathcal{L}^{\text{pe}}_{\mu,\kappa}(\tau)$, the class of positive
existential formulas, is defined similarly where we also allow $\vee$ and
$\bigvee_{j<\beta}$

  3. (3)

$\mathcal{L}^{\text{p}}_{\mu,\kappa}(\tau)$, the class of positive formulas,
is defined similarly where we allow $\vee$, $\bigvee_{j<\beta}$ and also the
universal quantifiers $\forall x$ and $\forall_{i<\alpha}v_{i}$.

  4. (4)

By a simple formula of $\mathcal{L}_{\mu,\kappa}(\tau)$, we mean a formula of
the form

$\phi=\exists_{i<\alpha}v_{i}[\bigwedge_{j<\beta}\phi_{j}],$

where each $\phi_{j}=\phi_{j}((v_{i})_{i<\alpha})$ is an atomic formula.

######  Lemma 2.9.

The following assertions are valid:

  1. (1)

Suppose ${\bf f}:{\mathbb{M}}\to{\mathbb{N}}$ is an $({\bf R},{\bf
S})$-bimodule homomorphism. Then ${\bf f}$ preserves
$\mathcal{L}_{\infty,\infty}^{\text{pe}}(\tau_{({\bf R},{\bf S})})$-formulas.

  2. (2)

Suppose in addition to 1) that $f$ is surjective. Then ${\bf f}$ preserves
$\mathcal{L}_{\infty,\infty}^{\text{p}}(\tau_{({\bf R},{\bf S})})$-formulas.

###### Proof.

(1). Suppose
$\phi((v_{i})_{i<\alpha})\in\mathcal{L}_{\infty,\infty}^{\text{pe}}(\tau_{({\bf
R},{\bf S})})$ and let $a_{i}\in{\mathbb{M}},$ where $i<\alpha.$ We need to
show:

${\mathbb{M}}\models\phi((a_{i})_{i<\alpha})\Rightarrow{\mathbb{N}}\models\phi(({\bf
f}(a_{i}))_{i<\alpha}).$

This can be proved by induction on the complexity of the formulas, and we
leave its routine check to the reader.

(2). This can be proved in a similar way; let us only consider the case of the
universal formula to show how the surjectivity of the function $\bf f$ is
used. Thus suppose that $\phi((v_{i})_{i<\alpha})=\forall
x\psi((v_{i})_{i<\alpha},x)$ is in
$\mathcal{L}_{\infty,\infty}^{\text{pe}}(\tau_{({\bf R},{\bf S})})$ and let
$a_{i}\in{\mathbb{M}},$ where $i<\alpha.$ We may assume that the lemma holds
for $\psi.$ Suppose ${\mathbb{M}}\models\phi((a_{i})_{i<\alpha})$. We show
that ${\mathbb{N}}\models\phi(({\bf f}(a_{i}))_{i<\alpha})$. Thus let
$b\in{\mathbb{N}}$. As $\bf f$ is surjective, there exists $a\in{\mathbb{M}}$
such that ${\bf f}(a)=b$ and by the assumption,
${\mathbb{M}}\models\psi((a_{i})_{i<\alpha},a)$. By the induction hypothesis,
${\mathbb{N}}\models\psi(({\bf f}(a_{i}))_{i<\alpha},{\bf f}(a))$, i.e.,
${\mathbb{N}}\models\psi(({\bf f}(a_{i}))_{i<\alpha},b)$. Since $b$ was
arbitrary we have ${\mathbb{N}}\models\phi(({\bf f}(a_{i}))_{i<\alpha})$. ∎

######  Remark 2.10.

Note that there is no need for ${\bf f}$ to preserve $\neg$ formulas. But if
${\bf f}$ is an isomorphism, then it preserves all formulas.

The next lemma shows that under suitable conditions, we can replace $\rm
cpe$-formulas by simple formulas.

######  Lemma 2.11.

Let $\tau$ be either $\tau_{{\bf R}}$ or $\tau_{({\bf R},{\bf S})}$, and
suppose $\mu_{1}\geq\mu,\kappa$ is regular. If
$\varphi(\bar{x})\in\mathcal{L}^{\rm cpe}_{\mu,\kappa}(\tau)$, then we can
find as equivalent simple formula in $\mathcal{L}_{\mu_{1},\mu_{1}}(\tau)$.

We are also interested in definable subsets of (bi)modules.

######  Definition 2.12.

Let $\tau$ be one of $\tau_{{\bf R}}$ or $\tau_{({\bf R},{\bf S})}$. Given a
$\tau$-structure ${\mathbb{M}}$ and a formula $\phi(x_{0},\cdots,x_{n-1})$ in
$\mathcal{L}_{\infty,\infty}(\tau)$, let

$\phi({\mathbb{M}})=\\{\langle
a_{0},\cdots,a_{n-1}\rangle\in~{}^{n}{{\mathbb{M}}}:M\models\phi(a_{0},\cdots,a_{n-1})\\}.$

Here, by $\mathop{\rm\ell g}(-)$ we mean the length function.

######  Lemma 2.13.

Let $\tau$ be either $\tau_{{\bf R}}$ or $\tau_{({\bf R},{\bf S})}$, and let
$\varphi(\bar{x})\in\mathcal{L}_{\mu,\kappa}^{\rm p}(\tau)$. Suppose
$\bar{z}_{\ell}\in{}^{\mathop{\rm\ell g}(\bar{x})}{\mathbb{M}}_{\ell}$, for
$\ell=1,2$ and $\bar{z}_{\ell}=\langle z^{\ell}_{i}:i<\mathop{\rm\ell
g}(\bar{x})\rangle$, ${\mathbb{M}}={\mathbb{M}}_{1}\oplus{\mathbb{M}}_{2}$,
and $\bar{z}=\langle z_{i}:i<\mathop{\rm\ell g}(\bar{x})\rangle$ where
${\mathbb{M}}\models z_{i}=z^{1}_{i}+z^{2}_{i}$. Then

${\mathbb{M}}\models\varphi(\bar{z})\quad\Leftrightarrow\quad\bigwedge\limits_{\ell=1}^{2}{\mathbb{M}}_{\ell}\models\varphi(\bar{z}_{\ell}).$

Furthermore, if for $\ell=1,2$ and $i<\mathop{\rm\ell
g}(\bar{x}),z^{\ell}_{i}=0_{M_{\ell}}$, then
${\mathbb{M}}_{\ell}\models\varphi(\bar{z}_{\ell})$ and
$\bar{z}=\bar{0}_{\mathop{\rm\ell g}(\bar{z})}$.

###### Proof.

The desired claim follows by an easy induction on the complexity of the
formula $\varphi(\bar{x})$. ∎

The above lemma implies if $\varphi(x)\in\mathcal{L}_{\mu,\kappa}^{\rm
cpe}(\tau),$ where $\tau$ is $\tau_{{\bf R}}$ or $\tau_{({\bf R},{\bf S})}$
and if ${\mathbb{M}}={\mathbb{M}}_{1}\oplus{\mathbb{M}}_{2}$, then
$\varphi({\mathbb{M}})=\varphi({\mathbb{M}}_{1})\oplus\varphi({\mathbb{M}}_{2})$.

######  Lemma 2.14.

For each bimodule ${\mathbb{M}}$, the following assertions hold:

  1. (1)

If $\varphi(x)\in\mathcal{L}^{{\rm cpe}}_{\infty,\infty}(\tau_{({\bf R},{\bf
S})})$, then $\varphi({\mathbb{M}})$ is a subgroup of ${\mathbb{M}}$.

  2. (2)

If $\varphi(x)\in\mathcal{L}^{{\rm cpe}}_{\infty,\infty}(\tau_{{\bf R}})$,
then $\varphi({\mathbb{M}})$ is a sub-right ${\bf S}$-module, but not
necessarily a sub-${\bf R}$-module. Furthermore, if ${\bf R}$ is commutative,
then it is a sub-${\bf R}$-module as well.

###### Proof.

This follows by induction on the complexity of $\varphi(x).$ The
straightforward details are leave to the reader. ∎

Another notion from model theory which is of importance for us is the notion
of omitting types:

######  Discussion 2.15.

Suppose $\tau$ is a countable languages and suppose $\mathcal{M}$ is a
$\tau$-structure.

  1. (1)

Given $A\subseteq\mathcal{M},$ by an $n$-type over $A$ we mean a set
$p(v_{1},\cdots,v_{n})$ of formulas, whose free variables are
$v_{1},\cdots,v_{n}$ such that every finite $p_{0}\subseteq p$ is realized,
i.e., there are $x_{1},\cdots,x_{n}$ in $\mathcal{M}$ such that for all
$\varphi\in p_{0},~{}\mathcal{M}\models\varphi(x_{1},\cdots,x_{n})$.

  2. (2)

A type is complete if it is maximal under inclusion. By the axiom of choice
each type can be extended into a complete type.

  3. (3)

The type $p$ is isolated by some formula $\psi(v_{1},\cdots,v_{n})\in p$ if
for every $\varphi\in p,$

$\mathcal{M}\models\forall x_{1}\cdots\forall x_{n}(\psi(x_{1},\cdots
x_{n})\rightarrow\varphi(x_{1},\cdots,x_{n})).$

It is clear that if $p$ is isolated by some formula
$\psi(v_{1},\cdots,v_{n})\in p$, then it is realized. The omitting types
theorem says that the converse is also true.

######  Lemma 2.16.

Let $\tau$ be a first order countable vocabulary and let $T$ be a complete
$\tau$-theory. If $p$ is a complete type which is not isolated, then there is
a countable $\tau$-structure $\mathcal{M}\models T$ which omits (i.e., does
not realize) $p$.

######  Definition 2.17.

Suppose that ${\mathbb{M}}_{0},{\mathbb{M}}_{1}$ and ${\mathbb{M}}_{2}$ are
(bi)modules and $\mathbb{g}_{i}:{\mathbb{M}}_{0}\to{\mathbb{M}}_{i}$ are
(bi)module homomorphisms for $i=1,2$. The amalgamation of ${\mathbb{M}}_{1}$
and ${\mathbb{M}}_{2}$ along ${\mathbb{M}}_{0}$ is
${\mathbb{M}}:=\frac{{\mathbb{M}}_{1}\oplus{\mathbb{M}}_{2}}{(\mathbb{g}_{1}(m),-\mathbb{g}_{2}(m):m\in{\mathbb{M}}_{0})}$.
There are natural maps $\mathbb{f}_{i}:{\mathbb{M}}_{i}\to{\mathbb{M}}$ such
that the following diagram is commutative:

${\mathbb{M}}_{0}$${\mathbb{M}}_{1}$${\mathbb{M}}$${\mathbb{M}}_{2}$

It may be nice to note that the common notation for this is
${\mathbb{M}}_{1}+_{{\mathbb{M}}_{0}}{\mathbb{M}}_{2}$.

######  Definition 2.18.

Let ${\mathbb{M}}$ be an ${\bf R}$-module and ${\bf S}$ be any subring of
${\rm End}_{\bf R}({\mathbb{M}})$. We say that ${\bf S}$ is algebraically
closed in ${\rm End}_{\bf R}({\mathbb{M}})$ if every finite system of
equations in several variables over ${\bf S}$ which has a solution in ${\rm
End}_{\bf R}({\mathbb{M}})$, also has a solution in ${\bf S}$

Let us recall the definition of pure semisimple rings.

######  Definition 2.19.

A ring ${\bf R}$ is left pure semisimple if every left ${\bf R}$-module is
pure-injective.

The next theorem gives several equivalent formulations for the above notion.

######  Theorem 2.20.

(See [33, Theorem 4.5.7]) The following conditions on a ring ${\bf R}$ are
equivalent:

  1. (a)

${\bf R}$ is left pure semisimple;

  2. (b)

every left ${\bf R}$-module is a direct sum of indecomposable modules;

  3. (c)

there is a cardinal number $\kappa$ such that every left ${\bf R}$-module is a
direct sum of modules, each of which is of cardinality less than $\kappa$;

  4. (d)

there is a cardinal number $\kappa$ such that every left ${\bf R}$-module is a
pure submodule of a direct sum of modules each of cardinality less than
$\kappa$.

The following result of Shelah gives a model theoretic criteria for rings
which are not pure semisimple, and plays an important role in this paper.

######  Theorem 2.21.

(Shelah, [36, 8.7]) Suppose ${\bf R}$ is not left pure semisimple. Then for
some bimodule ${\mathbb{M}}$ and a sequence
$\bar{\varphi}=\langle\varphi_{n}(x):n<\omega\rangle$, each $\varphi_{n}$ is a
conjunctive positive existential formula and the sequence
$\langle\varphi_{n}({\mathbb{M}}):n<\omega\rangle$ is strictly decreasing.

##  3\. Developing the framework of the construction

In this section we develop some part of the theory that we need for our
construction. The main result of this section is Theorem 3.44, which gives, in
ZFC, a semi-nice construction, that plays an essential role in the next
sections.

######  Definition 3.1.

  1. (1)

Let ${\mathfrak{E}}_{({\bf R},{\bf S})}={\mathfrak{E}}{({\bf R},{\bf S})}$ be
the class of all

${\mathfrak{e}}=\langle{\mathbb{N}}_{n},x_{n},g_{n}:n<\omega\rangle$

such that:

    1. (a)

${\mathbb{N}}_{n}$ is an $({\bf R},{\bf S})$-bimodule,

    2. (b)

$x_{n}\in{\mathbb{N}}_{n}$,

    3. (c)

$g_{n}$ is a bimodule homomorphism from ${\mathbb{N}}_{n}$ to
${\mathbb{N}}_{n+1}$ mapping $x_{n}$ to $x_{n+1}$.

Given ${\mathfrak{e}}\in{\mathfrak{E}}_{({\bf R},{\bf S})}$ we denote it by
${\mathfrak{e}}=\langle{\mathbb{N}}_{n}^{{\mathfrak{e}}},x_{n}^{{\mathfrak{e}}},g_{n}^{{\mathfrak{e}}}:n<\omega\rangle$.

  2. (2)

We call ${\mathfrak{e}}\in{\mathfrak{E}}_{({\bf R},{\bf S})}$ non-trivial if
for every $n$ there is no homomorphism from
${\mathbb{N}}^{\mathfrak{e}}_{n+1}$ to ${\mathbb{N}}^{\mathfrak{e}}_{n}$ as
${\bf R}$-modules mapping $x^{\mathfrak{e}}_{n+1}$ to $x^{\mathfrak{e}}_{n}$.

  3. (3)

Let
${\mathfrak{e}}=\langle{\mathbb{N}}_{n},x_{n},g_{n}:n<\omega\rangle\in{\mathfrak{E}}_{({\bf
R},{\bf S})}$. For each $n\leq m$, we define:

$g_{n,m}:=g_{n}\circ g_{n+1}\circ\ldots\circ g_{m-1}.$

We set $g_{n,n}:={\rm id}_{{\mathbb{N}}_{n}}$ and note that $n_{0}\leq
n_{1}\leq n_{2}$ implies $g_{n_{1},n_{2}}\circ
g_{n_{0},n_{1}}=g_{n_{0},n_{2}}$. We denote $g_{n,m}$ by
$g^{\mathfrak{e}}_{n,m}.$

  4. (4)

For ${\mathfrak{e}}\in{\mathfrak{E}}_{({\bf R},{\bf S})}$ and an infinite
${\mathcal{U}}\subseteq\omega$, let
${\mathfrak{e}}^{\prime}=:{\mathfrak{e}}\restriction{\mathcal{U}}$, the
restriction of ${\mathfrak{e}}$ to ${\mathcal{U}},$ be defined by

${\mathbb{N}}^{{\mathfrak{e}}^{\prime}}_{\ell}=:{\mathbb{N}}^{\mathfrak{e}}_{m(\ell)},\quad
x^{{\mathfrak{e}}^{\prime}}_{\ell}=:x^{\mathfrak{e}}_{m(\ell)},\quad
g^{{\mathfrak{e}}^{\prime}}_{\ell}=g^{\mathfrak{e}}_{m(\ell),m(\ell+1)},$

where $m(\ell)$ is the $\ell$–th member of $\mathcal{U}$. Clearly
${\mathfrak{e}}^{\prime}$ is in ${\mathfrak{E}}_{({\bf R},{\bf S})}$.

Let us define a special sub-class of ${\mathfrak{E}}_{({\bf R},{\bf S})}$.

######  Definition 3.2.

Let ${{\mathfrak{E}}}_{\mu,\kappa}$ be the class of
$\mathfrak{e}\in{\mathfrak{E}}_{({\bf R},{\bf S})}$ such that for each
$n<\omega$, ${\mathbb{N}}^{\mathfrak{e}}_{n}$, as a bimodule, is generated by
$<\kappa$ elements freely except satisfying $<\mu$ equations. This means,
there are $(x_{i})_{i<\alpha<\kappa}$ and atomic formulas $t_{j}=0,$ for
$j<\beta<\mu$, such that
${\mathbb{N}}^{\mathfrak{e}}_{n}=(\bigoplus_{i<\alpha}{\bf R}x_{i}{\bf
S})/{\mathbb{K}}$, where ${\mathbb{K}}$ is the bimodule generated by $\langle
t_{j}:j<\beta\rangle$.

We now define what it means for a sequence of formulas to be adequate with
respect to an element of ${\mathfrak{E}}_{({\bf R},{\bf S})}$:

######  Definition 3.3.

Let ${\mathfrak{e}}:=\langle{\mathbb{N}}_{n},x_{n},g_{n}:n<\omega\rangle$ and
$\bar{\varphi}:=\langle\varphi_{n}:n<\omega\rangle$.

  1. (1)

The sequence $\bar{\varphi}$ is called $(\mu,\kappa)$-adequate with respect to
${\mathfrak{e}}$ if the following conditions satisfied:

    1. $(\alpha)$

$\varphi_{n}=\varphi_{n}(x)$ is a formula from $\mathcal{L}^{\rm
cpe}_{\mu,\kappa}(\tau_{{\bf R}})$,

    2. $(\beta)$

$\varphi_{n+1}(x)\vdash\varphi_{n}(x)$ (for the class of ${\bf
R}$-modules)333we can restrict ourselves to those we actually use, i.e. the
ones from $\mathop{\mathrm{cl}}({\mathcal{K}})$.,

    3. $(\gamma)$

${\mathbb{N}}_{n}\models\varphi_{n}(x_{n})\ \&\ \neg\varphi_{n+1}(x_{n})$.

Also, we say ${\mathfrak{e}}$ is $(\mu,\kappa)$-adequate with respect to
${\mathfrak{e}}$.

  2. (2)

We say $\mathfrak{e}$ is explicitly $(\mu,\kappa$)-adequate with respect to
${\mathfrak{e}}$ if $\mathfrak{e}$ is $(\mu,\kappa)$-adequate with respect to
${\mathfrak{e}}$ and $\mathfrak{e}\in{\mathfrak{E}}_{\mu,\kappa}$.

  3. (3)

For simplicity, $\kappa$-adequate means $(\infty,\kappa)$-adequate with
respect to ${\mathfrak{e}}$. Also, adequate is referred to
$\aleph_{0}$-adequate with respect to ${\mathfrak{e}}$.

  4. (4)

We say that $\bar{\varphi}$ is $(\mu,\kappa)$-adequate if $\bar{\varphi}$ is
$(\mu,\kappa)$-adequate with respect to some
${\mathfrak{e}}\in{\mathfrak{E}}_{({\bf R},{\bf S})}$.

  5. (5)

We say $\bar{\varphi}$ is adequate with respect to
${\mathfrak{E}}\subseteq{\mathfrak{E}}_{({\bf R},{\bf S})}$ if it is adequate
with respect to some ${\mathfrak{e}}\in{\mathfrak{E}}$.

It follows from Lemma 2.9 that any $(\mu,\kappa)$-adequate
${\mathfrak{e}}\in{\mathfrak{E}}_{({\bf R},{\bf S})}$ is non-trivial.

######  Definition 3.4.

Let ${\mathfrak{e}}\in{{\mathfrak{E}}}$, $n<\omega$ and $\kappa$ be given. We
define

  1. (1)

$\varphi^{{\mathfrak{e}},\kappa}_{n}:=\bigwedge\left\\{\varphi(x):\varphi\in\mathcal{L}^{{\rm
cpe}}_{\infty,\kappa}(\tau_{{\bf R}})~{}{\rm
and~{}}{\mathbb{N}}^{\mathfrak{e}}_{n}\models\varphi(x^{\mathfrak{e}}_{n})\right\\}$444This
is not a formula being a conjunction on a class, not a set, but when we deal
with an ${\bf R}$-module ${\mathbb{M}}$ it is enough to restrict ourselves to
$\mathcal{L}^{{\rm cpe}}_{\lambda,\kappa}$ for
$\lambda=(||{\mathbb{M}}||^{<\kappa})^{+}$.,

  2. (2)

$\psi^{{\mathfrak{e}},\kappa}_{n}:=\bigwedge\left\\{\psi(x):\psi\in\mathcal{L}^{{\rm
cpe}}_{\infty,\kappa}(\tau_{({\bf R},{\bf S})})~{}{\rm
and~{}}{\mathbb{N}}^{\mathfrak{e}}_{n}\models\psi(x^{\mathfrak{e}}_{n})\right\\}$,

  3. (3)

$\bar{\varphi}^{{\mathfrak{e}},\kappa}:=\langle\varphi^{{\mathfrak{e}},\kappa}_{n}:n<\omega\rangle$,

  4. (4)

$\bar{\psi}^{{\mathfrak{e}},\kappa}:=\langle\psi^{{\mathfrak{e}},\kappa}_{n}:n<\omega\rangle,$

  5. (5)

$\varphi^{{\mathfrak{e}},\kappa}_{\omega}(x):=\bigwedge\limits_{n<\omega}\varphi^{{\mathfrak{e}},\kappa}_{n}(x)$,

  6. (6)

$\psi^{{\mathfrak{e}},\kappa}_{\omega}(x):=\bigwedge\limits_{n<\omega}\psi^{{\mathfrak{e}},\kappa}_{n}(x)$.

######  Remark 3.5.

In Definition 3.4, we omit the index $\kappa$, when it is
$\kappa(\mathfrak{e})$ for $\psi$ or $\kappa_{{\bf R}}(\mathfrak{e})$ for
$\varphi$, where $\kappa(\mathfrak{e})$ and $\kappa_{{\bf R}}(\mathfrak{e})$
are defined in Definition 3.7, see below.

The next lemma shows the relation between different
$\varphi^{{\mathfrak{e}},\kappa}_{n}$’s and
$\psi^{{\mathfrak{e}},\kappa}_{n}$’s.

######  Lemma 3.6.

For $n\geq m$,
$\varphi^{{\mathfrak{e}},\kappa}_{n}(x)\vdash\varphi^{{\mathfrak{e}},\kappa}_{m}(x)$,
$\psi^{{\mathfrak{e}},\kappa}_{n}(x)\vdash\psi^{{\mathfrak{e}},\kappa}_{m}(x)$
and
$\psi^{{\mathfrak{e}},\kappa}_{n}(x)\vdash\varphi^{{\mathfrak{e}},\kappa}_{n}(x)$.

###### Proof.

The lemma follows easily by applying Lemma 2.9 to $g^{\mathfrak{e}}_{m,n}$. ∎

######  Definition 3.7.

  1. (1)

For ${\mathfrak{e}}\in{\mathfrak{E}}_{{\bf R},{\bf S}}$ we define:

    1. (a)

$\kappa({\mathfrak{e}})$ as the first infinite cardinal $\kappa$ such that for
each $n<\omega$, the bimodule ${\mathbb{N}}^{{\mathfrak{e}}}_{n}$ is generated
by a set of $<\kappa$ elements.

    2. (b)

$\kappa_{{\bf R}}({\mathfrak{e}})$ be the first infinite cardinal $\kappa$
such that for each $n<\omega$, the bimodule
${\mathbb{N}}^{{\mathfrak{e}}}_{n}$ as an ${\bf R}$-module is generated by a
set of $<\kappa$ elements.

  2. (2)

For each ${\mathfrak{E}}\subseteq{\mathfrak{E}}_{({\bf R},{\bf S})}$, we let

    1. (a)

$\kappa({\mathfrak{E}}):=\mathop{\mathrm{sup}}\\{\kappa({\mathfrak{e}}):{\mathfrak{e}}\in{\mathfrak{E}}\\}$
and

    2. (b)

$\kappa({\mathfrak{E}})_{{\bf R}}:=\mathop{\mathrm{sup}}\\{\kappa_{{\bf
R}}({\mathfrak{e}}):{\mathfrak{e}}\in{\mathfrak{E}}\\}$.

We frequently use the following lemma.

######  Lemma 3.8.

  1. (1)

Assume ${\mathbb{N}}$ is a bimodule (resp. ${\bf R}$-module) generated by
$<\kappa$ members freely except satisfying $<\mu$ equations and
$x\in{\mathbb{N}}$. Then there is a simple formula
$\varphi\in\mathcal{L}_{\mu,\kappa}(\tau_{({\bf R},{\bf S})})$ (resp.
$\varphi\in\mathcal{L}_{\mu,\kappa}(\tau_{{\bf R}})$) such that for any
bimodule (resp. ${\bf R}$-module) ${\mathbb{M}}$ and $y\in{\mathbb{M}}$, the
following are equivalent:

    1. (a)

${\mathbb{M}}\models\varphi(y)$,

    2. (b)

for some homomorphism ${\bf f}$ from ${\mathbb{N}}$ into ${\mathbb{M}}$, ${\bf
f}(x)=y$ (if ${\mathbb{N}}$ is an ${\bf R}$-module this means ${\bf f}$ is an
${\bf R}$-homomorphism).

    3. (c)

for some ${\mathbb{N}}^{\prime}$ extending ${\mathbb{N}}$ and a homomorphism
${\bf f}$ from ${\mathbb{N}}^{\prime}$ into ${\mathbb{M}}$ we have ${\bf
f}(x)=y$.

  2. (2)

If ${\mathbb{N}}={\mathbb{N}}^{\prime}\oplus{\mathbb{N}}^{\prime\prime}$ and
$x\in N^{\prime}$, then it suffices that ${\mathbb{N}}^{\prime}$ is generated
by $<\kappa$ members freely except satisfying $<\mu$ equations.

  3. (3)

Clause (1) applies if ${\mathbb{N}}={\mathbb{N}}^{\mathfrak{e}}_{n}$,
$x=x_{n}^{\mathfrak{e}}$ and $\varphi=\psi^{{\mathfrak{e}},\kappa}_{n}$, when
${\mathbb{N}}^{\mathfrak{e}}_{n}$ is generated by $<\kappa$ members freely
except satisfying $<\mu$ equations (as a bimodule). Similarly for
$\varphi=\varphi^{{\mathfrak{e}},\kappa}_{n}$ for ${\bf R}$-modules.

###### Proof.

(1). We prove the lemma for the case of bimodules, exactly the same proof
works for ${\bf R}$-modules. Suppose ${\mathbb{N}}$ is a bimodule generated by
$<\kappa$ members freely except satisfying $<\mu$ equations and
$x\in{\mathbb{N}}$. So it has the form ${\mathbb{N}}=\bigoplus_{i<\alpha}{\bf
R}x_{i}{\bf S}/{\mathbb{K}}$, where ${\mathbb{K}}$ is the bimodule generated
by $\langle t_{j}:j<\beta\rangle$, where each $t_{j}=0$ is an atomic formula,
$\alpha<\kappa$ and $\beta<\mu$. Then

  * •

$x=\sum_{i<d}\sum_{h<h_{d}}r_{i,h}x_{i}s_{i,h}$, where $d,h_{d}<\omega,$
$r_{i,h}\in{\bf R}$ and $s_{i,h}\in{\bf S}$. We should remark that for all but
finitely many of them we have $r_{i,h}x_{i}s_{i,h}=0.$

  * •

$t_{j}=\sum_{i<d}\sum_{h<h_{d}}r^{j}_{i,h}x_{i}s^{j}_{i,h},$ where
$r^{j}_{i,h}\in{\bf R}$ and $s^{j}_{i,h}\in{\bf S}$. Again, for all but
finitely many of them we have $r^{j}_{i,h}x_{i}s^{j}_{i,h}=0.$

Now consider the formula

$\varphi(v)=\exists_{i<\alpha}v_{i}[\bigwedge_{j<\beta}\sum_{i<\alpha}\sum_{h<h_{d}}r^{j}_{i,h}v_{i}s^{j}_{i,h}=0\wedge
v=\sum_{i<\alpha}\sum_{h<h_{d}}r_{i,h}v_{i}s_{i,h}].$

It is clearly a simple formula in $\mathcal{L}_{\mu,\kappa}(\tau_{({\bf
R},{\bf S})})$. Now suppose ${\mathbb{M}}\models\varphi(y)$. Thus we can find
$y_{i},$ for $i<\alpha$, such that in $M$,

$\sum_{i<\alpha}\sum_{h<h_{d}}r^{j}_{i,h}y_{i}s^{j}_{i,h}=0,$

and

$y=\sum_{i<\alpha}\sum_{h<h_{d}}r_{i,h}y_{i}s_{i,h}.$

Define ${\bf f}:{\mathbb{N}}\to{\mathbb{M}}$ so that for each
$i<\alpha,f(x_{i})=y_{i}.$ Then $f$ is as required.

(2). This is clear and (3) follows from the fact that
$\psi^{{\mathfrak{e}},\kappa}_{n}\vdash\varphi$ (because
${\mathbb{N}}^{\mathfrak{e}}_{n}\models\varphi(x_{n}^{\mathfrak{e}})$). ∎

The next lemma shows that each non-trivial
${\mathfrak{e}}\in{{\mathfrak{E}}}_{\mu,\kappa}$ has a canonical adequate
sequence.

######  Lemma 3.9.

If ${\mathfrak{e}}\in{{\mathfrak{E}}}_{\mu,\kappa}$ is non-trivial and
$\mu\geq\kappa\geq\aleph_{0}$ then $\bar{\varphi}^{{\mathfrak{e}},\kappa}$ is
$(\mu,\kappa)$-adequate with respect to $\mathfrak{e}$.

###### Proof.

It is easily seen, by the structure of each ${\mathbb{N}}_{n}^{\mathfrak{e}}$
that $\varphi^{\mathfrak{e},\kappa}_{n}$ is equivalent to a formula of
$\mathcal{L}^{{\rm cpe}}_{\infty,\kappa}(\tau_{{\bf R}})$. According to Lemma
3.6
$\varphi^{\mathfrak{e},\kappa}_{n+1}\vdash\varphi^{\mathfrak{e},\kappa}_{n}.$
By the definition of $\varphi^{\mathfrak{e},\kappa}_{n}$, we have
${\mathbb{N}}_{n}^{\mathfrak{e}}\models\varphi^{\mathfrak{e},\kappa}_{n}(x_{n}^{\mathfrak{e}})$.
It remains to show that
${\mathbb{N}}_{n}^{\mathfrak{e}}\models\neg\varphi^{\mathfrak{e},\kappa}_{n+1}(x_{n}^{\mathfrak{e}})$.
Suppose not. It follows from Lemma 3.8(3) that there is a bimodule
homomorphism ${\bf
h}:{\mathbb{N}}_{n+1}^{\mathfrak{e}}\to{\mathbb{N}}_{n}^{\mathfrak{e}}$ with
${\bf h}(x_{n+1}^{\mathfrak{e}})=x_{n}^{\mathfrak{e}},$ contradicting the non-
triviality of $\mathfrak{e}$. ∎

######  Definition 3.10.

A $\lambda$-_context_ is a tuple

${\mathfrak{m}}:=({\mathcal{K}},{\mathbb{M}}_{\ast},{\mathfrak{E}},{\bf
R},{\bf S},{\bf
T})=({\mathcal{K}}^{\mathfrak{m}},{\mathbb{M}}_{\ast}^{\mathfrak{m}},{\mathfrak{E}}^{\mathfrak{m}},{\bf
R}^{\mathfrak{m}},{\bf S}^{\mathfrak{m}},{\bf T}^{\mathfrak{m}})$

where

  1. (1)

${\bf R},{\bf S}$ and ${\bf T}$ are as usual.

  2. (2)

${\mathcal{K}}$ is a set of $({\bf R},{\bf S})$-bimodules.

  3. (3)

${{\mathfrak{E}}}$ is a subset of ${{\mathfrak{E}}}_{\infty,\lambda}$ closed
under restrictions (i.e., if ${\mathfrak{e}}\in{{\mathfrak{E}}}$ and
${\mathcal{U}}\subseteq\omega$ is infinite then
${\mathfrak{e}}\restriction{\mathcal{U}}\in{{\mathfrak{E}}}$).

  4. (4)

If ${\mathfrak{e}}\in{{\mathfrak{E}}}$, then $\
{\mathbb{N}}^{\mathfrak{e}}_{n}\in{\mathcal{K}}$ for each $n<\omega$.

  5. (5)

${\mathbb{M}}_{\ast}$ is a bimodule. If ${\mathbb{M}}_{\ast}$ is omitted we
mean the zero bimodule.

Let $\chi({\mathcal{K}})$ be the minimal cardinal $\chi\geq\aleph_{0}$ such
that every ${\mathbb{N}}\in{\mathcal{K}}$ is generated, as a bimodule, by a
set of $<\chi$ members and set

$||{\mathfrak{m}}||=\sum\\{\|{\mathbb{N}}\|:{\mathbb{N}}\in{\mathcal{K}}\\}+\|{\mathfrak{E}}\|+||{\mathbb{M}}_{\ast}||||{\bf
R}||+||{\bf S}||+\aleph_{0}.$

######  Remark 3.11.

We omit $\lambda$ from the $\lambda$-context, if it is clear from the context
and then we mean for the minimal such $\lambda$.

######  Convention 3.12.

From now on suppose that
${{\mathfrak{m}}}=({\mathcal{K}},{\mathbb{M}}_{\ast},{{\mathfrak{E}}},{\bf
R},{\bf S},{\bf T})$ is a context.

######  Definition 3.13.

  1. (1)

Given a context
${\mathfrak{m}}=({\mathcal{K}},{\mathbb{M}}_{\ast},{\mathfrak{E}},{\bf R},{\bf
S},{\bf T})$, we set:

     * •

$c\ell_{\rm is}({\mathcal{K}})$ be the class of bimodules isomorphic to some
${\mathbb{K}}\in{\mathcal{K}}$.

     * •

$c\ell^{\kappa}_{\rm is}({\mathcal{K}})$ be the class of bimodules of the form
${\mathbb{M}}=\bigoplus\limits_{i<j}{\mathbb{M}}_{i}$, where $j<\kappa$, and
for $i<j,{\mathbb{M}}_{i}\in c\ell_{\rm is}({\mathcal{K}})$.

     * •

$c\ell({\mathcal{K}})=c\ell_{\rm ds}({\mathcal{K}})$ be the class of bimodules
isomorphic to a direct sum of bimodules from $c\ell_{\rm is}({\mathcal{K}})$
(so $c\ell_{\rm is}(c\ell({\mathcal{K}}))=c\ell({\mathcal{K}})$).

     * •

$\kappa({\mathfrak{m}})=\kappa({\mathfrak{E}}^{\mathfrak{m}})$.

     * •

$\kappa_{\bf R}({\mathfrak{m}})=\kappa_{\bf
R}({\mathfrak{E}}^{\mathfrak{m}})$.

  2. (2)

Following Definition 3.3, let us say that $\bar{\varphi}$ is adequate for a
context ${\mathfrak{m}}$ if it is adequate with respect to
${\mathfrak{E}}^{\mathfrak{m}}$.

######  Definition 3.14.

By a ${\mathcal{K}}$-bimodule we mean a bimodule from $c\ell_{\rm
is}({\mathcal{K}})$. Also, we say ${\mathbb{M}}_{1}$ is a
${\mathcal{K}}$-direct summand of ${\mathbb{M}}_{2}$ if
${\mathbb{M}}_{2}={\mathbb{M}}_{1}\oplus{\mathbb{K}}$ for some
${\mathbb{K}}\in c\ell_{\rm ds}({\mathcal{K}})$.

######  Definition 3.15.

We say ${\mathbb{M}}_{1}$ is an almost direct ${\mathcal{K}}$-summand of
${\mathbb{M}}_{2}$ with respect to $\kappa$ denoted ${\mathbb{M}}_{1}\leq^{\rm
ads}_{{\mathcal{K}},\kappa}{\mathbb{M}}_{2}$, if player II has a winning
strategy in the following game
$\Game_{{\mathcal{K}},\kappa}^{{\mathbb{M}}_{1},{\mathbb{M}}_{2}}$ of length
$\omega$: in the $n^{\rm th}$ move player I chooses a subset
$A_{n}\subseteq{\mathbb{M}}_{2}$ of cardinality $<\kappa$, and then player II
chooses a sub-bimodule ${\mathbb{K}}_{n}\subseteq{\mathbb{M}}_{2}.$ Player II
wins iff:

  1. (a)

$A_{n}\subseteq{\mathbb{M}}_{1}+\sum\limits_{\ell\leq n}{\mathbb{K}}_{\ell}$,

  2. (b)

${\mathbb{K}}_{n}$ is in $c\ell^{\kappa}_{\rm is}({\mathcal{K}})$,

  3. (c)

${\mathbb{M}}_{1}+\sum\limits_{\ell<\omega}{\mathbb{K}}_{\ell}={\mathbb{M}}_{1}\oplus\bigoplus\limits_{\ell<\omega}{\mathbb{K}}_{\ell}$.

We usually write $\leq_{\kappa}$ or $\leq^{\rm ads}_{\kappa}$ instead of
$\leq^{\rm ads}_{{\mathcal{K}},\kappa}$ if ${\mathcal{K}}$ is clear. We also
may write $\leq_{{{\mathfrak{m}}},\kappa}$ or $\leq^{\rm
ads}_{{{\mathfrak{m}}},\kappa}$, when
$\mathcal{K}=\mathcal{K}^{{\mathfrak{m}}}$.

We now define another order between bimodules and then connect it to the above
defined notion.

######  Definition 3.16.

Let ${\mathbb{M}}_{1}$ and ${\mathbb{M}}_{2}$ be two bimodules.

  1. (1)

${\mathbb{M}}_{1}\leq^{\rm pr}_{\varphi}{\mathbb{M}}_{2}$ means that
${\mathbb{M}}_{1}\subseteq{\mathbb{M}}_{2}$ and if $\psi=\psi(\bar{x})$ is a
sub-formula of $\varphi$ and $\bar{z}\in{}^{\mathop{\rm\ell
g}(\bar{x})}{\mathbb{M}}_{1}$ then

${\mathbb{M}}_{1}\models\psi(\bar{z})\iff{\mathbb{M}}_{2}\models\psi(\bar{z}).$

  2. (2)

By ${\mathbb{M}}_{1}\leq^{\rm qr}_{\varphi}{\mathbb{M}}_{2}$ we mean that
${\mathbb{M}}_{1}\subseteq{\mathbb{M}}_{2}$ and if $\psi=\psi(\bar{x})$ is a
sub-formula of $\varphi$ and $\bar{z}\in{}^{(\mathop{\rm\ell
g}\bar{x})}{\mathbb{M}}_{1}$, then

${\mathbb{M}}_{2}\models\varphi(\bar{z})\Rightarrow{\mathbb{M}}_{1}\models\psi(\bar{z}).$

  3. (3)

Assume $\mathcal{F}$ is a set of formulas. The notation
${\mathbb{M}}_{1}\leq^{\rm pr}_{\mathcal{F}}{\mathbb{M}}_{2}$ (resp.
${\mathbb{M}}_{1}\leq^{\rm qr}_{\mathcal{F}}{\mathbb{M}}_{2}$) means
${\mathbb{M}}_{1}\leq^{\rm pr}_{\varphi}{\mathbb{M}}_{2}$ (resp.
${\mathbb{M}}_{1}\leq^{\rm qr}_{\varphi}{\mathbb{M}}_{2}$) for all
$\varphi\in\mathcal{F}.$ In the special case
$\bar{\varphi}:=\langle\varphi_{n}:n<\omega\rangle$, the property
${\mathbb{M}}_{1}\leq^{\rm pr}_{\bar{\varphi}}{\mathbb{M}}_{2}$ holds iff
${\mathbb{M}}_{1}\leq^{\rm pr}_{\varphi_{n}}{\mathbb{M}}_{2}$ for each $n$.

For instance, let $\varphi$ be the following simple formula

$\varphi=\varphi(x)=(\exists
y_{0},\ldots,y_{i}\ldots)_{i<\alpha}\bigwedge\limits_{j<\beta}[a_{j}x=\sum\limits_{i<\alpha}b_{j,i}y_{i}]$

where $a_{j},b_{j,i}\in{\bf R}$. Then ${\mathbb{M}}_{1}\leq^{\rm
pr}_{\varphi}{\mathbb{M}}_{2}$ means that if for some $x\in{\mathbb{M}}_{1}$
and $y_{i}\in{\mathbb{M}}_{2}$ we have

$a_{j}x-\sum_{i<\alpha}b_{j,i}y_{i}=0\quad\forall j<\beta$

then there are $y^{\prime}_{i}\in{\mathbb{M}}_{1}$ such that

$a_{j}x-\sum\limits_{i<\alpha}b_{j,i}y^{\prime}_{i}=0\quad\forall j<\beta.$

######  Remark 3.17.

Note that if $\varphi$ is existential (e.g. a simple formula), then $\leq^{\rm
qr}_{\varphi}$ is equal to $\leq^{{\mathord{\mathrm{pr}}}}_{\varphi}$.

The next lemma connects the above defined relations to each other.

######  Lemma 3.18.

Let ${\mathbb{M}}_{1}$ and ${\mathbb{M}}_{2}$ be two bimodules such that
${\mathbb{M}}_{1}$ is ${\mathcal{K}}$-nice. The following assertions are true:

  1. (1)

${\mathbb{M}}_{1}\leq_{\kappa}{\mathbb{M}}_{2}$ iff ${\mathbb{M}}_{1}\leq^{\rm
pr}_{\mathcal{L}_{\infty,\kappa}(\tau_{({\bf R},{\bf S})})}{\mathbb{M}}_{2}$.

  2. (2)

If ${\mathbb{M}}_{1}\leq_{\kappa}{\mathbb{M}}_{2}$ and
$\varphi(x)\in\mathcal{L}^{\rm cpe}_{\infty,\kappa}(\tau_{({\bf R},{\bf
S})})$, then
$\varphi({\mathbb{M}}_{1})={\mathbb{M}}_{1}\cap\varphi({\mathbb{M}}_{2})$.

###### Proof.

Clause (2) follows from (1) and the definition of $\leq^{\rm pr}$, so let us
prove (1). Let ${\mathbb{M}}_{0}$ witness ${\mathbb{M}}_{1}$ is
$\mathcal{K}$-nice.

Suppose ${\mathbb{M}}_{1}\leq_{\kappa}{\mathbb{M}}_{2}$. We prove, by
induction on the complexity of the formula
$\varphi\in\mathcal{L}_{\infty,\kappa}(\tau_{({\bf R},{\bf S})}),$ that
${\mathbb{M}}_{1}\leq^{\rm pr}_{\varphi}{\mathbb{M}}_{2}$.

The only non-trivial case where the assumption
${\mathbb{M}}_{1}\leq_{\kappa}{\mathbb{M}}_{2}$ is used is the case of
existential quantifier555The case of universal quantifier then follows easily
as $\forall\equiv\neg\exists\neg.$, so let us suppose
$\varphi(v_{i})_{i<\alpha}$ is of the form

$\exists_{j<\beta}w_{j}[\psi((w_{j})_{j<\beta},(v_{i})_{i<\alpha})],$

the claim holds for $\psi=\psi((w_{j})_{j<\beta},(v_{i})_{i<\alpha})$ and
suppose $(x_{i})_{i<\alpha}\in{\mathbb{M}}_{1}$. We want to show that

${\mathbb{M}}_{1}\models\varphi((x_{i})_{i<\alpha})\iff{\mathbb{M}}_{2}\models\varphi((x_{i})_{i<\alpha}).$

As ${\mathbb{M}}_{1}$ is a submodule of ${\mathbb{M}}_{2},$ by the induction
hypothesis, if ${\mathbb{M}}_{1}\models\varphi((x_{i})_{i<\alpha})$, then
${\mathbb{M}}_{2}\models\varphi((x_{i})_{i<\alpha}).$ Conversely suppose that
${\mathbb{M}}_{2}\models\varphi((x_{i})_{i<\alpha}).$ Then we can find
$(y_{j})_{j<\beta}\in{\mathbb{M}}_{2}$ such that
${\mathbb{M}}_{2}\models\psi((y_{j})_{j<\beta},(x_{i})_{i<\alpha}).$

Now consider the game
$\Game_{\mathcal{K},\kappa}^{{\mathbb{M}}_{1},{\mathbb{M}}_{2}}$, in which
player I plays $A_{n}=\\{y_{j}:j<\beta\\}$ at each step $n$. At the end, we
have bimodules $\\{{\mathbb{K}}_{n}:n<\omega\\}$ such that:

  * •

$A_{n}\subseteq{\mathbb{M}}_{1}+\sum\limits_{\ell\leq n}{\mathbb{K}}_{\ell}$,

  * •

${\mathbb{K}}_{n}$ is in $c\ell^{\kappa}_{\rm is}({\mathcal{K}})$,

  * •

${\mathbb{M}}_{1}+\sum\limits_{\ell<\omega}{\mathbb{K}}_{\ell}={\mathbb{M}}_{1}\oplus\bigoplus\limits_{\ell<\omega}{\mathbb{K}}_{\ell}$.

Set ${\mathbb{K}}=\bigoplus\limits_{\ell<\omega}{\mathbb{K}}_{\ell}.$ It then
follows that
${\mathbb{M}}_{1}\oplus{\mathbb{K}}\models\varphi((x_{i})_{i<\alpha})$. In
view of Lemma 2.13, ${\mathbb{M}}_{1}\models\varphi((x_{i})_{i<\alpha}).$

Conversely, suppose that ${\mathbb{M}}_{1}\leq^{\rm
pr}_{\mathcal{L}_{\infty,\kappa}(\tau_{({\bf R},{\bf S})})}{\mathbb{M}}_{2}$.
Let us first state a few simple facts:

  1. (i)

Suppose $\alpha<\kappa$ and $\bar{b}\in$${}^{\alpha}{\mathbb{M}}_{2}$. Then
there exists $\bar{a}\in$${}^{\alpha}{\mathbb{M}}_{1}$ such that

$({\mathbb{M}}_{1},\bar{a})\equiv_{\mathcal{L}_{\infty,\kappa}(\tau_{({\bf
R},{\bf S})})}({\mathbb{M}}_{2},\bar{b}).$

To see this, let $\alpha<\kappa$ and
$\bar{b}\in$${}^{\alpha}{\mathbb{M}}_{2}$. Suppose by contradiction that there
is no $\bar{a}\in{}^{\alpha}{\mathbb{M}}_{1}$ such that
$({\mathbb{M}}_{1},\bar{a})\equiv_{\mathcal{L}_{\infty,\kappa}(\tau_{({\bf
R},{\bf S})})}({\mathbb{M}}_{2},\bar{b}).$ Thus for each
$\bar{a}\in{}^{\alpha}{\mathbb{M}}_{1}$ we can find
$\phi_{\bar{a}}(\bar{\nu})\in\mathcal{L}_{\infty,\kappa}(\tau_{({\bf R},{\bf
S})})$ such that

${\mathbb{M}}_{1}\models\neg\phi_{\bar{a}}(\bar{a})~{}~{}\&~{}~{}{\mathbb{M}}_{2}\models\phi_{\bar{a}}(\bar{b}).$

Let
$\phi(\bar{\nu})=\bigwedge\\{\phi_{\bar{a}}(\bar{\nu}):\bar{a}\in{}^{\alpha}{\mathbb{M}}_{1}\\}\in\mathcal{L}_{\infty,\kappa}(\tau_{({\bf
R},{\bf S})}).$ Then
${\mathbb{M}}_{1}\models~{}\forall\bar{\nu}\neg\phi(\bar{\nu})$, while
${\mathbb{M}}_{2}\models\phi(\bar{b})$, a contradiction to our assumption.

  2. (ii)

Suppose $\bar{a}_{1}\in{}^{\mathop{\rm\ell
g}(\bar{a}_{1})}{\mathbb{M}}_{1},\bar{a}_{2}\in{}^{\mathop{\rm\ell
g}(\bar{a}_{2})}{\mathbb{M}}_{1}$ with $\bar{a}_{1}\unlhd\bar{a}_{2}$ and
$\bar{b}_{1}\in{}^{\mathop{\rm\ell g}(\bar{a}_{1})}{\mathbb{M}}_{2}$ are such
that
$({\mathbb{M}}_{1},\bar{a}_{1})\equiv_{\mathcal{L}_{\infty,\kappa}(\tau_{({\bf
R},{\bf S})})}({\mathbb{M}}_{2},\bar{b}_{1}).$ There there is some
$\bar{b}_{2}\in{}^{\mathop{\rm\ell g}(\bar{a}_{2})}{\mathbb{M}}_{2}$ such that
$\bar{b}_{1}\unlhd\bar{b}_{2}$ and
$({\mathbb{M}}_{1},\bar{a}_{2})\equiv_{\mathcal{L}_{\infty,\kappa}(\tau_{({\bf
R},{\bf S})})}({\mathbb{M}}_{2},\bar{b}_{2}).$

  3. (iii)

Suppose $\alpha<\kappa,\bar{a}\in{}^{\alpha}{\mathbb{M}}_{1}$,
$\bar{b}\in{}^{\alpha}{\mathbb{M}}_{2}$ and
$({\mathbb{M}}_{1},\bar{a})\equiv_{\mathcal{L}_{\infty,\kappa}(\tau_{({\bf
R},{\bf S})})}({\mathbb{M}}_{2},\bar{b})$. Let $\bar{c}=\langle
b_{i}-a_{i}:i<\alpha\rangle$ and let ${\mathbb{K}}$ be the submodule of
${\mathbb{M}}_{2}$ generated by $\bar{c}$. Then
$\bar{b}\in{\mathbb{M}}_{1}+{\mathbb{K}}$ and
${\mathbb{M}}_{1}+{\mathbb{K}}={\mathbb{M}}_{1}\oplus{\mathbb{K}}.$

  4. (iv)

Suppose $\langle K_{\ell}:\ell<\omega\rangle$ is an increasing sequence of
submodules of ${\mathbb{M}}_{2}$ such that for each
$\ell<\omega,{\mathbb{M}}_{1}+{\mathbb{K}}_{\ell}={\mathbb{M}}_{1}\oplus{\mathbb{K}}_{\ell}$.
Then
${\mathbb{M}}_{1}+\sum\limits_{\ell<\omega}{\mathbb{K}}_{\ell}={\mathbb{M}}_{1}\oplus\bigoplus\limits_{\ell<\omega}{\mathbb{K}}_{\ell}$.

We are now ready to define a winning strategy for the game
$\Game_{{\mathcal{K}},\kappa}^{{\mathbb{M}}_{1},{\mathbb{M}}_{2}}$. Fix a
well-ordering $<^{*}$ of ${\mathbb{M}}_{2}$. To start set $A_{-1}=\emptyset$
and $\bar{b}_{-1}=\langle\rangle.$ At stage $n$, suppose player I has chosen
$A_{n}\subseteq{\mathbb{M}}_{2}$ of size $<\kappa.$ We may assume that
$A_{n}\supseteq A_{n-1}.$ Let $\bar{a}_{n}=\langle a_{n}(i):i<\mathop{\rm\ell
g}(\bar{a}_{n})\rangle$ enumerate $A_{n}$ in the $<^{*}$-order in such a way
that $A_{n-1}$ is an initial segment of the enumeration. By (i) and (ii),
choose $\bar{b}_{n}=\langle b_{n}(i):i<\mathop{\rm\ell
g}(\bar{a}_{n})\rangle\in{\mathbb{M}}_{2}$ such that
$\bar{b}_{n}\unrhd\bar{b}_{n-1}$ and
$({\mathbb{M}}_{1},\bar{a}_{n})\equiv_{\mathcal{L}_{\infty,\kappa}(\tau_{({\bf
R},{\bf S})})}({\mathbb{M}}_{2},\bar{b}_{n}).$ Let ${\mathbb{K}}_{n}$ be the
submodule of ${\mathbb{M}}_{2}$ generated by
$\\{b_{n}(i)-a_{n}(i):i<\mathop{\rm\ell g}(\bar{a}_{n})\\}.$ Now (iii) and
(iv) guarantee that items (a) and (c) of Definition 3.15 are satisfied. This
completes the proof. ∎

######  Claim 3.19.

Let $\varphi$ be in $\mathcal{L}^{\rm cpe}_{\infty,\infty}(\tau_{({\bf R},{\bf
S})})$. The following holds:

  1. (1)

Assuming $\varphi=\varphi(\bar{x})$ we can find a simple formula
$\varphi_{\ast}(\bar{z})\in\mathcal{L}_{\infty,\infty}(\tau_{({\bf R},{\bf
S})})$ satisfying:

    1. (a)

if
${\mathbb{M}}_{1}\leq^{\rm{\mathord{\mathrm{pr}}}}_{\varphi_{\ast}}{\mathbb{M}}_{2}$
and $\bar{x}\in{\mathbb{M}}_{2}$ then $\langle
x_{i}+{\mathbb{M}}_{1}:i<\mathop{\rm\ell
g}(\bar{x})\rangle\in\varphi({\mathbb{M}}_{2}/{\mathbb{M}}_{1})$ iff
$\bar{x}\in\varphi({\mathbb{M}}_{2})+{\mathbb{M}}_{1}$.

    2. (b)

if $\varphi(\bar{x})\in\mathcal{L}^{\rm cpe}_{\mu,\mu}(\tau_{({\bf R},{\bf
S})})$ and $\mu$ is regular, then
$\varphi_{\ast}(\bar{z})\in\mathcal{L}_{\mu,\mu}(\tau_{({\bf R},{\bf S})})$.

  2. (2)

Assume ${\mathbb{M}}_{2}\in c\ell_{\rm ds}({\mathcal{K}})$ and
${\mathbb{M}}_{1}$ is ${\mathcal{K}}$-nice. If
${\mathbb{M}}_{1}\oplus{\mathbb{M}}_{2}={\mathbb{M}}_{3}$, then
${\mathbb{M}}_{1}\leq^{\rm pr}_{\varphi}{\mathbb{M}}_{3}$.

  3. (3)

If $\langle{\mathbb{M}}_{i}:i<\delta\rangle$ is increasing,
${\mathbb{M}}_{i}\leq^{\rm pr}_{\varphi}{\mathbb{M}}$ for all $i<\delta$ and
$\varphi\in\mathcal{L}_{\infty,\mathop{\mathrm{cf}}(\delta)}(\tau_{({\bf
R},{\bf S})})$, then $\bigcup\limits_{i<\delta}{\mathbb{M}}_{i}\leq^{\rm
pr}_{\varphi}{\mathbb{M}}$ and ${\mathbb{M}}_{j}\leq^{\rm
pr}_{\varphi}\bigcup\limits_{i<\delta}{\mathbb{M}}_{i}$ for all $j<\delta$.

  4. (4)

Assume $\varphi$ is a simple formula of
$\mathcal{L}_{\kappa,\kappa}(\tau_{({\bf R},{\bf S})})$. Then
${\mathbb{M}}_{1}<^{\rm pr}_{\varphi}{\mathbb{M}}_{2}$ iff for every
$<\kappa$-generated submodule ${\mathbb{N}}\subseteq{\mathbb{M}}_{2}$ we have
${\mathbb{M}}_{1}\leq^{\rm pr}_{\varphi}{\mathbb{M}}_{1}+{\mathbb{N}}$.

  5. (5)

If ${\mathbb{M}}=\bigoplus\limits_{t\in I}{\mathbb{M}}_{t}$ and
$\varphi(x)\in\mathcal{L}_{\infty,\infty}^{p}(\tau_{({\bf R},{\bf S})})$, then
$\varphi({\mathbb{M}})=\bigoplus\limits_{t\in I}\varphi({\mathbb{M}}_{t})$.

###### Proof.

(1). For simplicity, suppose that $\varphi=\varphi(x)$. By Lemma 2.11, we can
assume the formula $\varphi$ is simple, so let it be of the form

$\varphi=\exists_{i<\alpha}y_{i}\bigwedge_{j<\beta}\varphi_{j}(\bar{y},x)$

where each $\varphi_{j}(\bar{y},x)$ is an atomic formula. But the only atomic
relation is equality, so by moving to one side, without loss of generality
$\varphi_{j}(\bar{y},x)$ is of the form

$\sigma_{j}(\bar{y},x)=0$

for some term $\sigma_{i}$. Let $\bar{z}=\langle z_{j}:j<\beta\rangle$ and

$\varphi_{\ast}(\bar{z})=(\exists
x,\bar{y})[\bigwedge_{j<\beta}\sigma_{j}(\bar{y},x)=z_{j}].$

We show that $\varphi_{\ast}(\bar{z})$ is as required. Clause (b) clearly
holds. In order to prove clause (a), take
${\mathbb{M}}_{1}\leq^{\rm{\mathord{\mathrm{pr}}}}_{\varphi_{\ast}}{\mathbb{M}}_{2}$
and $x\in{\mathbb{M}}_{2}$. First, assume that
$x+{\mathbb{M}}_{1}\in\varphi({\mathbb{M}}_{2}/{\mathbb{M}}_{1})$. Then

${\mathbb{M}}_{2}/{\mathbb{M}}_{1}\models\exists\bar{y}\bigwedge_{j<\beta}\sigma_{j}(\bar{y},x+{\mathbb{M}}_{1})=0.$

We can find $\bar{b}=(b_{i})_{i<\alpha}\in{\mathbb{M}}_{2}$ such that
${\mathbb{M}}_{2}/{\mathbb{M}}_{1}\models~{}\sigma_{j}(\bar{b}+{\mathbb{M}}_{1},x+{\mathbb{M}}_{1})=0$
for all $j<\beta$. 666By $\bar{b}+{\mathbb{M}}_{1}$ we mean $\langle
b_{i}+{\mathbb{M}}_{1}:i<\alpha\rangle$. By definition,
$\sigma_{j}(\bar{b}+{\mathbb{M}}_{1},x+{\mathbb{M}}_{1})=c_{j}+{\mathbb{M}}_{1}$
for some $c_{j}\in M_{1}$. We set $\bar{c}:=\langle c_{j}:j<\beta\rangle$.
Then ${\mathbb{M}}_{2}\models\varphi_{*}(\bar{c})$. Hence,

$M_{2}\models\sigma_{j}(\bar{b},x)=c_{j}$

for all $j<\beta$. We apply our assumption to see
${\mathbb{M}}_{1}\models\varphi_{*}(\bar{c})$, e.g., there are
$\bar{b}^{\prime}=(b^{\prime}_{i})_{i<\alpha}$ and $x^{\prime}$ in
${\mathbb{M}}_{1}$ such that

$M_{1}\models\sigma_{j}(\bar{b}^{\prime},x^{\prime})=c_{j}$

for all $j<\beta$. Then ${\mathbb{M}}_{2}$ satisfies the same formulas. In
particular,

$M_{2}\models\sigma_{j}((b_{i}-b^{\prime}_{i})_{i<\alpha},x-x^{\prime})=0,$

which implies $M_{2}\models\varphi(x-x^{\prime}).$ Thus
$x\in\varphi({\mathbb{M}}_{2})+{\mathbb{M}}_{1}$.

Conversely, suppose that $x\in\varphi({\mathbb{M}}_{2})+{\mathbb{M}}_{1}.$ Let
$y\in{\mathbb{M}}_{1}$ be such that $x-y\in\varphi({\mathbb{M}}_{2})$. Then
${\mathbb{M}}_{2}\models\varphi(x-y)$, and consequently there is
$\bar{b}=((b_{i})_{i<\alpha})\in{\mathbb{M}}_{2}$ such that

$M_{2}\models\sigma_{j}(\bar{b},x-y)=0$

for all $j<\beta$. Set $\bar{z}:=\bar{0}.$ Then
${\mathbb{M}}_{2}\models\varphi_{*}(\bar{z})$, and hence by our assumption
${\mathbb{M}}_{1}\models\varphi_{*}(\bar{z})$. It follows that for some
$\bar{b}^{\prime}=(b^{\prime}_{i})_{i<\alpha}$ and $x^{\prime}$ in
${\mathbb{M}}_{1}$ and

$M_{1}\models\sigma_{j}(\bar{b}^{\prime},x^{\prime})=0$

for all $j<\beta$. Thus, ${\mathbb{M}}_{2}$ satisfies the same formulas.
Therefore,

$M_{2}\models\sigma_{j}((b_{i}-b^{\prime}_{i})_{i<\alpha},x-y-x^{\prime})=0$

for all $j<\beta.$ Since $\bar{b}^{\prime},y+x^{\prime}$ are in
${\mathbb{M}}_{1}$, we have

${\mathbb{M}}_{2}/{\mathbb{M}}_{1}\models\sigma_{j}(\bar{b}+{\mathbb{M}}_{1},x+{\mathbb{M}}_{1})=0$

for all $j<\beta$. This implies
${\mathbb{M}}_{2}/{\mathbb{M}}_{1}\models\varphi(x+{\mathbb{M}}_{1})$. Hence,
$x+{\mathbb{M}}_{1}\in\varphi({\mathbb{M}}_{2}/{\mathbb{M}}_{1})$.

(2). This is in Lemma 2.13.

(3). Let us first show that $\bigcup_{i<\delta}{\mathbb{M}}_{i}\leq^{\rm
pr}_{\varphi}{\mathbb{M}}$. To see this, let $\psi(\bar{x})$ be a subformula
of $\varphi$ and $\bar{z}\in\bigcup_{i<\delta}{\mathbb{M}}_{i}.$ There is
$i_{*}<\delta$ such that $\bar{z}\in{\mathbb{M}}_{i_{*}}$. Then for all
$i_{*}\leq i<\delta$, ${\mathbb{M}}_{i}\models\psi(\bar{z})$ iff
${\mathbb{M}}\models\psi(\bar{z})$. It follows that

$\bigcup_{i<\delta}{\mathbb{M}}_{i}\models\psi(\bar{z})\iff{\mathbb{M}}\models\psi(\bar{z}).$

Let $j<\delta$, $\psi(\bar{x})$ a subformula of $\varphi$ and
$\bar{z}\in{\mathbb{M}}_{j}$. We conclude from the above argument that

${\mathbb{M}}_{j}\models\psi(\bar{z})\iff{\mathbb{M}}\models\psi(\bar{z})\iff\bigcup_{i<\delta}{\mathbb{M}}_{i}\models\psi(\bar{z}).$

Thus ${\mathbb{M}}_{j}\leq^{\rm
pr}_{\varphi}\bigcup_{i<\delta}{\mathbb{M}}_{i}$.

(4) It is easily seen that if ${\mathbb{M}}_{1}\leq^{\rm
pr}_{\varphi}{\mathbb{M}}_{2}$ then ${\mathbb{M}}_{1}\leq^{\rm
pr}_{\varphi}{\mathbb{M}}_{1}+{\mathbb{N}}$ for each $<\kappa$-generated
submodule ${\mathbb{N}}$ of ${\mathbb{M}}_{2}$.

Conversely, suppose that the above condition holds. Let $\psi(\bar{x})$ be a
subformula of $\varphi$. Then it is a simple formula as well, thus it is of
the form

$\exists_{i<\alpha}v_{i}\bigwedge_{j<\beta}[\sigma_{j}(\bar{x},\bar{v})=0],$

for some terms $\sigma_{j}.$ Let $\bar{x}\in{\mathbb{M}}_{1}$. It is obvious
that if ${\mathbb{M}}_{1}\models\psi(\bar{x})$, then
${\mathbb{M}}_{2}\models\psi(\bar{x})$. Now let
${\mathbb{M}}_{2}\models\psi(\bar{x})$. Thus we can find
$\bar{b}\in{\mathbb{M}}_{2}$ such that

${\mathbb{M}}_{2}\models\bigwedge_{j<\beta}[\sigma_{j}(\bar{x},\bar{b})=0].$

Let ${\mathbb{N}}$ be the submodule generated by $\bar{b}$. Then
${\mathbb{M}}_{1}+{\mathbb{N}}\models\bigwedge_{j<\beta}[\sigma_{j}(\bar{x},\bar{b})=0]$.
We combine this along with our assumption to conclude that
${\mathbb{M}}_{1}\models\psi(\bar{x})$.

(5). We proceed by induction on the complexity of the formula $\varphi$ (see
also Lemma 2.13), and we leave the routine check to the reader. ∎

Let $\kappa$ be an infinite cardinal and let
${\mathfrak{m}}=({\mathcal{K}},{\mathbb{M}}_{\ast},{{\mathfrak{E}}},{\bf
R},{\bf S},{\bf T})$ be a context. Set ${\rm{\rm
Mod}}_{{\mathfrak{m}},\kappa}:=\\{{\mathbb{M}}:{\mathbb{M}}_{\ast}\leq^{\rm
ads}_{{\mathcal{K}},\kappa}{\mathbb{M}}\\}$.

######  Lemma 3.20.

Let ${\mathfrak{m}}$ be a context as above and let $\kappa$ be an infinite
cardinal.

  1. (1)

If ${\mathbb{M}}_{1}\leq_{\kappa}{\mathbb{M}}_{2},$ then ${\mathbb{M}}_{1}$ is
a sub-module of ${\mathbb{M}}_{2}.$

  2. (2)

$\leq_{\kappa}$ is a partial order on ${\rm{\rm
Mod}}_{{\mathfrak{m}},\kappa}.$

  3. (3)

If $\kappa\leq\theta$ and ${\mathbb{M}}\leq_{\theta}{\mathbb{N}}$ then
${\mathbb{M}}\leq_{\kappa}{\mathbb{N}}$.

  4. (4)

If $\langle{\mathbb{M}}_{i}:i<\delta\rangle$ is a $\leq_{\kappa}$-increasing
and continuous in ${\rm{\rm Mod}}_{{\mathfrak{m}},\kappa}$, then

    1. (a)

$\bigcup\limits_{i<\delta}{\mathbb{M}}_{i}\in{\rm{\rm
Mod}}_{{\mathfrak{m}},\kappa},$

    2. (b)

For each
$j<\delta,{\mathbb{M}}_{j}\leq_{\kappa}\bigcup\limits_{i<\delta}{\mathbb{M}}_{i}$.

  5. (5)

If ${\mathbb{M}}_{1},{\mathbb{M}}_{2},{\mathbb{M}}_{3}\in{\rm{\rm
Mod}}_{{\mathfrak{m}},\kappa}$ are such that
${\mathbb{M}}_{1},{\mathbb{M}}_{2}\leq_{\kappa}{\mathbb{M}}_{3}$ and
${\mathbb{M}}_{1}\subseteq{\mathbb{M}}_{2}$, then
${\mathbb{M}}_{1}\leq_{\kappa}{\mathbb{M}}_{2}$.

  6. (6)

If ${\mathbb{M}}_{1}\leq_{\kappa}{\mathbb{M}}_{2}$ and
$A\subseteq{\mathbb{M}}_{2}$ has cardinality $<\kappa$ then for some
${\mathbb{N}}\in c\ell^{\kappa}_{is}({\mathcal{K}})$, we have
$A\subseteq{\mathbb{M}}_{1}+{\mathbb{N}}={\mathbb{M}}_{1}\oplus{\mathbb{N}}\leq_{\kappa}{\mathbb{M}}_{2}$.

  7. (7)

The pair $({\rm Mod}_{{\mathfrak{m}},\kappa},\leq_{\kappa})$ has the
amalgamation property.

  8. (8)

There is a cardinal $\chi$ such that if
${\mathbb{M}}_{1}\subseteq{\mathbb{N}}$ are in ${\rm{\rm
Mod}}_{{\mathfrak{m}},\kappa}$, then there is ${\mathbb{M}}_{2}\in{\rm{\rm
Mod}}_{{\mathfrak{m}},\kappa}$ such that
${\mathbb{M}}_{1}\subseteq{\mathbb{M}}_{2}\leq_{\kappa}{\mathbb{N}}$ and
$|{\mathbb{M}}_{2}|\leq|{\mathbb{M}}_{1}|+\chi$.

###### Proof.

The lemma follows easily from Lemmas 3.18 and 3.19. ∎

######  Definition 3.21.

We say the bimodule ${\mathbb{N}}$ is free as an ${\bf S}$–module if

  1. (a)

As an ${\bf R}$-module, it can be written as
$\mathop{{\bigoplus}}\limits_{i<\alpha}{\mathbb{N}}_{i}$ where each
${\mathbb{N}}_{i}$ is a sub-${\bf R}$-module of ${\mathbb{N}}$.

  2. (b)

If ${\mathbb{M}}$ is a bimodule, $i<\alpha$ and
$g:{\mathbb{N}}_{i}\longrightarrow{\mathbb{M}}$ is an ${\bf R}$-homomorphism,
then there is a unique bimodule homomorphism
$h:{\mathbb{N}}\longrightarrow{\mathbb{M}}$ extending $g$.

The next lemma can be proved as in Lemma 3.8.

######  Lemma 3.22.

Let ${\mathfrak{e}}\in{{\mathfrak{E}}}$.

  1. (1)

There exists a formula
$\varphi^{\mathfrak{e}}_{n}\in\mathcal{L}_{\infty,\infty}^{cpe}(\tau_{{\bf
R}})$ such that for an ${\bf R}$-module ${\mathbb{M}}$,
${\mathbb{M}}\models\varphi^{\mathfrak{e}}_{n}(x)$ iff for some
${\mathbb{M}}^{\prime}$ with
${\mathbb{M}}\leq_{\aleph_{0}}{\mathbb{M}}^{\prime}$,
${\mathbb{M}}^{\prime}\models\varphi^{{\mathfrak{e}},\infty}_{n}(x)$;
equivalently there is an ${\bf R}$-module homomorphism from
${\mathbb{N}}^{\mathfrak{e}}_{n}$ into ${\mathbb{M}}^{\prime}$ mapping
$x^{\mathfrak{e}}_{n}$ to $x$.

  2. (2)

There exists $\psi^{\mathfrak{e}}_{n}\in\mathcal{L}_{\infty,\infty}(\tau_{{\bf
R}})$ such that for a bimodule ${\mathbb{M}}$,
${\mathbb{M}}\models\psi^{\mathfrak{e}}_{n}(x)$ iff for some
${\mathbb{M}}^{\prime}$ with
${\mathbb{M}}\leq_{\aleph_{0}}{\mathbb{M}}^{\prime}$,
${\mathbb{M}}^{\prime}\models\psi^{{\mathfrak{e}},\infty}_{n}(x)$.

The above lemma suggests the following:

######  Definition 3.23.

By $\varphi^{\mathfrak{e}}_{\omega}(x)$ we mean
$\bigwedge\limits_{n<\omega}\varphi^{\mathfrak{e}}_{n}(x)$ and
$\overline{\varphi}^{\mathfrak{e}}:=\langle\varphi^{\mathfrak{e}}_{n}:n<\omega\rangle$.
Similarly, we define
$\psi^{\mathfrak{e}}_{\omega}(x):=\bigwedge\limits_{n<\omega}\psi^{\mathfrak{e}}_{n}(x)$
and let
$\overline{\psi}^{\mathfrak{e}}:=\langle\psi^{\mathfrak{e}}_{n}:n<\omega\rangle$.

######  Definition 3.24.

  1. (1)

A bimodule ${\mathbb{M}}$ is called ${{\mathfrak{E}}}$-closed if for every
${\mathfrak{e}}\in{{\mathfrak{E}}}$, $n<\omega$ and every $x\in{\mathbb{M}}$
we have

${\mathbb{M}}\models\psi^{\mathfrak{e}}_{n}(x)\iff{\mathbb{M}}\models\psi^{{\mathfrak{e}},\infty}_{n}(x).$

  2. (2)

We say that $\mathfrak{e}$ is non-trivial for ${{\mathfrak{m}}}$ if for every
$n<\omega$ there exists ${\mathbb{M}}\in{\mathcal{K}}^{\mathfrak{m}}$ such
that ${\mathbb{M}}_{*}^{\mathfrak{m}}\oplus{\mathbb{M}}\models(\exists
x)[\varphi^{\mathfrak{e}}_{n}(x)\wedge\neg\varphi^{\mathfrak{e}}_{\omega}(x)]$.

######  Fact 3.25.

It is easily seen that if ${\mathfrak{e}}\in{\mathfrak{E}}^{{\mathfrak{m}}}$
is adequate and each ${\mathbb{N}}^{\mathfrak{e}}_{n}$ is finitely generated
(as an ${\bf R}$-module), then ${\mathfrak{e}}$ is non-trivial for
${{\mathfrak{m}}}$.

######  Convention 3.26.

From now on we fix a context
${\mathfrak{m}}=({\mathcal{K}},{\mathbb{M}}_{*},{\mathfrak{E}},{\bf R},{\bf
S},{\bf T})$ which is non-trivial which means:

  1. (a)

${\mathcal{K}}\neq\emptyset$,

  2. (b)

${\mathfrak{E}}\neq\emptyset$,

  3. (c)

every ${\mathfrak{e}}\in{\mathfrak{E}}_{\infty,\kappa}$ is non-trivial for
${\mathfrak{m}}$.

######  Definition 3.27.

Let ${\mathfrak{e}}\in{{\mathfrak{E}}}$ and $\bar{\varphi}$ is adequate with
respect to $\mathfrak{e}$.

  1. (1)

Suppose ${\mathbb{M}}$ is a bimodule and ${\bf h_{1}},{\bf
h_{2}}:{\mathbb{N}}^{\mathfrak{e}}_{n}\to{\mathbb{M}}$ are bimodule
homomorphisms. We define

${\mathbb{L}}^{{\mathfrak{e}},\bar{\varphi},{\bf h_{1}},{\bf
h_{2}}}_{n}({\mathbb{M}}):=\\{z\in\varphi_{n}({\mathbb{N}}^{\mathfrak{e}}_{n}):{\bf
h_{1}}(z)={\bf h_{2}}(z)\ \mod\ \varphi_{\omega}({\mathbb{M}})\\}.$

For simplicity, and if there is no danger of confusion, we set

${\mathbb{L}}^{{\mathfrak{e}},\bar{\varphi},{\bf h_{1}},{\bf
h_{2}}}_{n}:={\mathbb{L}}^{{\mathfrak{e}},\bar{\varphi},{\bf h_{1}},{\bf
h_{2}}}_{n}({\mathbb{M}}).$

If $\bar{\varphi}=\bar{\varphi}^{{\mathfrak{e}},\kappa}$ we may write
${\mathbb{L}}^{{\mathfrak{e}},\kappa,{\bf h_{1}},{\bf h_{2}}}_{n}$ and if
$\kappa=\aleph_{0}$ we may omit it.

  2. (2)

Let
${\mathbb{L}}^{\mathfrak{e}}_{n}[{\mathfrak{m}}]:=\bigcap\big{\\{}{\mathbb{L}}^{{\mathfrak{e}},{\bf
h_{1}},{\bf h_{2}}}_{n}:\text{~{}for
some~{}}{\mathbb{M}}\in{\mathcal{K}}\cup\\{{\mathbb{M}}_{\ast}\\},{\bf
h_{1}},{\bf h_{2}}:{\mathbb{N}}^{\mathfrak{e}}_{n}\to{\mathbb{M}}\text{~{}are
bimodule }$ $\text{homomorphisms and~{}}{\bf h_{1}}(x_{n}^{\mathfrak{e}})={\bf
h_{2}}(x_{n}^{\mathfrak{e}})\big{\\}}.$ We may write
${\mathbb{L}}^{\mathfrak{e}}_{n}:={\mathbb{L}}^{\mathfrak{e}}_{n}[{\mathfrak{m}}]$,
when ${\mathfrak{m}}$ is clear from the context.

  3. (3)

${\mathbb{L}}^{\mathfrak{e}}_{n}[\mathcal{K}]$, is defined similarly but we
only require ${\mathbb{M}}\in{\mathcal{K}}$.

######  Lemma 3.28.

Let ${\mathfrak{e}}\in{{\mathfrak{E}}}$. Then, there are bimodules
${\mathbb{P}}^{\mathfrak{e}}$, ${\mathbb{P}}^{\mathfrak{e}}_{n}$ and
${\mathbb{K}}_{n}$, embeddings ${\bf
h}^{\mathfrak{e}}_{n}:{\mathbb{N}}^{\mathfrak{e}}_{n}\longrightarrow{\mathbb{P}}^{\mathfrak{e}}$,
an element $x=x_{\mathfrak{e}}$, an embedding ${\bf
f}^{\mathfrak{e}}:{\mathbb{N}}^{\mathfrak{e}}_{0}\longrightarrow{\mathbb{P}}^{\mathfrak{e}}$
and $x_{n}:=x_{{\mathfrak{e}},n}\in{\mathbb{P}}^{\mathfrak{e}}$ such that:

  1. (a)

$\mathop{\mathrm{Rang}}({\bf h}^{\mathfrak{e}}_{n})\cap\sum_{m\neq
n}\mathop{\mathrm{Rang}}({\bf h}^{\mathfrak{e}}_{m})=\\{0\\}$,

  2. (b)

For each ${\mathbb{K}}_{n}$ we have:

${\mathbb{P}}^{\mathfrak{e}}=\left(\sum_{\ell<n}\mathop{\mathrm{Rang}}({\bf
h}^{\mathfrak{e}}_{\ell})\right)+{\mathbb{K}}_{n}={\bigoplus}_{\ell<n}\mathop{\mathrm{Rang}}({\bf
h}^{\mathfrak{e}}_{\ell})\oplus{\mathbb{K}}_{n},$

and ${\mathbb{K}}_{n}$ is a direct sum of copies of
${\mathbb{N}}^{\mathfrak{e}}_{m}$’s.

  3. (c)

$\sum_{n<\omega}\mathop{\mathrm{Rang}}({\bf h}^{\mathfrak{e}}_{n})$ is not a
direct summand of ${\mathbb{P}}^{\mathfrak{e}}$; moreover, for some embeddings
${\bf
f}^{\mathfrak{e}}_{n}:{\mathbb{N}}^{\mathfrak{e}}_{n}\longrightarrow{\mathbb{P}}^{\mathfrak{e}}$
satisfying:

    1. $(\ast)_{1}$

$x_{n}=\sum_{\ell<n}{\bf
h}^{\mathfrak{e}}_{\ell}\big{(}x^{\mathfrak{e}}_{\ell}\big{)}\in\sum_{\ell<n}\mathop{\mathrm{Rang}}(h^{\mathfrak{e}}_{\ell})$,

    2. $(\ast)_{2}$

$x-x_{n}\in\varphi^{\mathfrak{e}}_{n}({\mathbb{P}}^{\mathfrak{e}})$,

    3. $(\ast)_{3}$

${\bf f}^{\mathfrak{e}}={\bf f}^{\mathfrak{e}}_{0}$,

    4. $(\ast)_{4}$

${\bf f}^{\mathfrak{e}}\big{(}x^{\mathfrak{e}}_{0}\big{)}=x$,

    5. $(\ast)_{5}$

$x\notin\sum_{n<\omega}\mathop{\mathrm{Rang}}({\bf
h}^{\mathfrak{e}}_{n})+\varphi^{\mathfrak{e}}_{\omega}({\mathbb{P}}^{\mathfrak{e}})$.

  4. (d)

${\mathbb{P}}^{\mathfrak{e}}$ is the direct sum of the copies
$\mathop{\mathrm{Rang}}({\bf f}^{\mathfrak{e}}_{n})$ of
${\mathbb{N}}_{n}^{\mathfrak{e}}$.

###### Proof.

Set
${\mathbb{P}}^{\mathfrak{e}}:=\mathop{{\bigoplus}}\limits_{n<\omega}{\mathbb{N}}^{\mathfrak{e}}_{n}$
and denote the natural embedding from ${\mathbb{N}}^{\mathfrak{e}}_{n}$ into
${\mathbb{P}}^{\mathfrak{e}}$ by ${\bf f}^{\mathfrak{e}}_{n}$. In particular,

${\mathbb{P}}^{\mathfrak{e}}=\mathop{{\bigoplus}}\limits_{n<\omega}\mathop{\mathrm{Rang}}({\bf
f}^{\mathfrak{e}}_{n}),$

i.e., (d) holds. We define ${\bf
h}^{\mathfrak{e}}_{n}:{\mathbb{N}}^{\mathfrak{e}}_{n}\longrightarrow{\mathbb{M}}$,
for all $n<\omega$, as follows, where $g^{\mathfrak{e}}_{n,n+1}$ is defined as
in Definition 3.1(3):

${\bf h}^{\mathfrak{e}}_{n}(y)={\bf f}^{\mathfrak{e}}_{n}(y)-{\bf
f}^{\mathfrak{e}}_{n+1}(g^{\mathfrak{e}}_{n,n+1}(y))\mbox{ for every
}y\in{\mathbb{N}}^{\mathfrak{e}}_{n}.$

As ${\mathbb{P}}^{\mathfrak{e}}=\mathop{\mathrm{Rang}}({\bf
f}^{\mathfrak{e}}_{n})\oplus\mathop{{\bigoplus}}\limits_{\ell\neq
n}\mathop{\mathrm{Rang}}({\bf f}^{\mathfrak{e}}_{\ell})$, we observe that
${\bf h}^{\mathfrak{e}}_{n}$ is a bimodule embedding. Set also

${\mathbb{P}}^{\mathfrak{e}}_{n}:=\sum\limits_{\ell<n}\mathop{\mathrm{Rang}}({\bf
h}^{\mathfrak{e}}_{\ell}).$

The next claim can be proved easily.

######  Claim 3.29.

Adopt the above notation. Then, for each $n$,

$\mathop{\mathrm{Rang}}({\bf
f}^{\mathfrak{e}}_{n})\oplus\mathop{\mathrm{Rang}}({\bf
f}^{\mathfrak{e}}_{n+1})=\mathop{\mathrm{Rang}}({\bf
h}^{\mathfrak{e}}_{n})\oplus\mathop{\mathrm{Rang}}({\bf
f}^{\mathfrak{e}}_{n+1}).$

In particular,
${\mathbb{P}}^{\mathfrak{e}}=\mathop{{\bigoplus}}\limits_{\ell<n}\mathop{\mathrm{Rang}}({\bf
h}^{\mathfrak{e}}_{n})\oplus\mathop{{\bigoplus}}\limits_{\ell\geq
n}\mathop{\mathrm{Rang}}({\bf f}^{\mathfrak{e}}_{\ell})$ for all $n$.

We set ${\mathbb{K}}_{n}:=\mathop{{\bigoplus}}\limits_{\ell\geq
n}\mathop{\mathrm{Rang}}({\bf f}^{\mathfrak{e}}_{\ell})$. In view of Claim
3.29, the items $(a)$, $(b)$ and $(d)$ of Lemma 3.28 are hold. Next we shall
show that $x:={\bf f}^{\mathfrak{e}}_{0}((x^{\mathfrak{e}}_{0}))$ and
$x_{n}:=\sum\limits_{\ell<n}{\bf
h}^{\mathfrak{e}}_{\ell}(x^{\mathfrak{e}}_{\ell})$ are as required in Lemma
3.28(c). This implies the first statement of $(*)_{1}$ in item (c). Trivially
$(*)_{3}$ and $(\ast)_{4}$ are true. Now,

$\begin{array}[]{ll}x&={\bf f}^{\mathfrak{e}}_{0}(x^{\mathfrak{e}}_{0})\\\
&={\bf h}^{\mathfrak{e}}_{0}(x^{\mathfrak{e}}_{0})+{\bf
f}^{\mathfrak{e}}_{1}(g^{\mathfrak{e}}_{0,1}(x^{\mathfrak{e}}_{0}))\\\ &={\bf
h}^{\mathfrak{e}}_{0}(x^{\mathfrak{e}}_{0})+{\bf
f}^{\mathfrak{e}}_{1}(x^{\mathfrak{e}}_{1})\\\ &={\bf
h}^{\mathfrak{e}}_{0}(x^{\mathfrak{e}}_{0})+{\bf
h}^{\mathfrak{e}}_{1}(x^{\mathfrak{e}}_{1})+{\bf
f}^{\mathfrak{e}}_{2}(x^{\mathfrak{e}}_{2}).\end{array}$

By induction on $n$ we have

$x=\sum\limits_{\ell<n}{\bf
h}^{\mathfrak{e}}_{\ell}(x^{\mathfrak{e}}_{\ell})+{\bf
f}^{\mathfrak{e}}_{n}(x^{\mathfrak{e}}_{n})=x_{n}+{\bf
f}^{\mathfrak{e}}_{n}(x^{\mathfrak{e}}_{n}).$

Clearly,
$x_{n}\in\mathop{{\bigoplus}}\limits_{\ell<n+1}\mathop{\mathrm{Rang}}({\bf
h}^{\mathfrak{e}}_{\ell})\subseteq\mathop{{\bigoplus}}\limits_{\ell<\omega}\mathop{\mathrm{Rang}}({\bf
h}^{\mathfrak{e}}_{\ell})$ and by the choice of ${\bf f}^{\mathfrak{e}}_{n}$
the second term is in
$\varphi^{\mathfrak{e}}_{n}({\mathbb{P}}^{\mathfrak{e}})$, so
$(*)_{1}+(*)_{2}$ of clause (c) holds. We shall now show that
$x\notin\sum\limits_{\ell<\omega}\mathop{\mathrm{Rang}}({\bf
h}^{\mathfrak{e}}_{\ell})+\varphi^{\mathfrak{e}}_{\omega}({\mathbb{P}}^{\mathfrak{e}})$,
i.e., $(*)_{5}$ holds.

Suppose by contradiction that there is an $n\geq 1$ such that $x\in
y+\sum\limits_{\ell<n}\mathop{\mathrm{Rang}}({\bf h}^{\mathfrak{e}}_{\ell})$,
where $y\in\varphi^{\mathfrak{e}}_{\omega}({\mathbb{P}}^{\mathfrak{e}})$.

We now define a bimodule endomorphism ${\bf f}_{n}\in{\rm
End}({\mathbb{P}}^{\mathfrak{e}})$. As
${\mathbb{P}}^{\mathfrak{e}}=\mathop{{\bigoplus}}\limits_{\ell<\omega}\mathop{\mathrm{Rang}}({\bf
f}^{\mathfrak{e}}_{\ell})$, it is clearly enough to define each ${\bf
f}_{n}\restriction\mathop{\mathrm{Rang}}(f^{\mathfrak{e}}_{\ell})$, for $n\geq
1,$ separately. Recall that ${\bf f}^{\mathfrak{e}}_{\ell}$ is one-to-one. Let
$z\in{\mathbb{N}}^{\mathfrak{e}}_{\ell}$. We define

${\bf f}_{n}({\bf f}^{\mathfrak{e}}_{\ell}(z))=\left\\{\begin{array}[]{ll}{\bf
f}^{\mathfrak{e}}_{n}(g^{\mathfrak{e}}_{\ell,n}(z))&\mbox{if }\ell\leq n,\\\
0&\mbox{otherwise }\end{array}\right.$

Now, we bring the following claim:

######  Claim 3.30.

  1. 1)

If $\ell\not=n$, then ${\bf f}_{n}\restriction\mathop{\mathrm{Rang}}({\bf
h}^{\mathfrak{e}}_{\ell})$ is identically zero.

  2. 2)

If $\ell=n$, then ${\bf f}_{n}(x)={\bf
f}^{\mathfrak{e}}_{n}(x^{\mathfrak{e}}_{n}).$

###### Proof.

1): For $\ell>n$ this is trivial, so suppose that $\ell<n$. Let
$z\in{\mathbb{N}}^{\mathfrak{e}}_{\ell}$. Then ${\bf
h}^{\mathfrak{e}}_{\ell}(z)={\bf f}^{\mathfrak{e}}_{\ell}(z)-{\bf
f}^{\mathfrak{e}}_{{\ell}+1}(g^{\mathfrak{e}}_{{\ell},{\ell}+1}(z))$. So,

$\begin{array}[]{ll}{\bf f}_{n}({\bf h}^{\mathfrak{e}}_{\ell}(z))&={\bf
f}_{n}\left({\bf f}^{\mathfrak{e}}_{\ell}(z)-{\bf
f}^{\mathfrak{e}}_{{\ell}+1}(g^{\mathfrak{e}}_{{\ell},{\ell}+1}(z))\right)\\\
&={\bf f}_{n}({\bf f}^{\mathfrak{e}}_{\ell}(z))-{\bf f}_{n}({\bf
f}^{\mathfrak{e}}_{{\ell}+1}(g^{\mathfrak{e}}_{{\ell},{\ell}+1}(z)))\\\ &={\bf
f}^{\mathfrak{e}}_{n}(g^{\mathfrak{e}}_{\ell,n}(z))-{\bf
f}^{\mathfrak{e}}_{n}(g^{\mathfrak{e}}_{\ell,\ell+1}(g^{\mathfrak{e}}_{\ell+1,n}(z)))\\\
&={\bf f}^{\mathfrak{e}}_{n}(g^{\mathfrak{e}}_{\ell,n}(z))-{\bf
f}^{\mathfrak{e}}_{n}(g^{\mathfrak{e}}_{\ell,n}(z))\\\ &=0.\end{array}$

The third equation holds because of clause (e) of Lemma 3.28.

2): It is enough to recall ${\bf f}_{n}(x)={\bf f}_{n}({\bf
f}^{\mathfrak{e}}_{0}(x^{\mathfrak{e}}_{0}))={\bf
f}^{\mathfrak{e}}_{n}(g^{\mathfrak{e}}_{0,n}({\bf
f}^{\mathfrak{e}}_{0}(x^{\mathfrak{e}}_{0}))={\bf
f}^{\mathfrak{e}}_{n}(x^{\mathfrak{e}}_{n}).$ ∎

Since
$x^{\mathfrak{e}}_{n}\notin\varphi_{\omega}({\mathbb{N}}^{\mathfrak{e}}_{n})$,
we have ${\bf f}_{n}(x)={\bf f}^{\mathfrak{e}}_{n}(x^{\mathfrak{e}}_{n})\neq
0$. Indeed, we have

$x^{\mathfrak{e}}_{n}\notin\varphi^{\mathfrak{e}}_{\omega}(\mathop{\mathrm{Rang}}({\bf
f}^{\mathfrak{e}}_{n}))=\varphi^{\mathfrak{e}}_{\omega}({\mathbb{P}}^{\mathfrak{e}})\cap\mathop{\mathrm{Rang}}({\bf
f}^{\mathfrak{e}}_{n}).$

But $x-y\in\sum\limits_{\ell<n}\mathop{\mathrm{Rang}}({\bf
h}^{\mathfrak{e}}_{\ell})$ and ${\bf
f}_{n}{\restriction}\mathop{\mathrm{Rang}}(h^{\mathfrak{e}}_{n})$ is zero, so
${\bf f}_{n}(x-y)=0$. Hence ${\bf f}_{n}(x)={\bf f}_{n}(y)$. However ${\bf
f}_{n}(x)\notin\varphi^{\mathfrak{e}}_{\omega}({\mathbb{P}}^{\mathfrak{e}})$.
Thus, ${\bf
f}_{n}(y)\notin\varphi^{\mathfrak{e}}_{\omega}({\mathbb{P}}^{\mathfrak{e}})$.
Recall that
$y\in\varphi^{\mathfrak{e}}_{\omega}({\mathbb{P}}^{\mathfrak{e}})$. Therefore,
${\bf
f}_{n}(y)\in\varphi^{\mathfrak{e}}_{\omega}({\mathbb{P}}^{\mathfrak{e}})$.
This contradiction completes the proof of Lemma 3.28. ∎

######  Lemma 3.31.

  1. (1)

Let $x\in{\mathbb{M}}_{1}$, ${\mathfrak{e}}\in{\mathfrak{E}}$ and
$\varphi\in\\{\tilde{\varphi}^{\mathfrak{e}}_{\alpha},\tilde{\psi}^{\mathfrak{e}}_{\alpha}:\alpha\leq\omega\\}$.
If ${\mathbb{M}}_{1}\leq_{\aleph_{0}}{\mathbb{M}}_{2}$, then

${\mathbb{M}}_{1}\models\varphi(x)\iff{\mathbb{M}}_{2}\models\varphi(x).$

  2. (2)

If ${\mathbb{M}}_{\ast}\leq_{\aleph_{0}}{\mathbb{M}}_{1}$ and
${{\mathfrak{E}}}^{\prime}\subseteq{{\mathfrak{E}}}$, then there is an
${\mathbb{M}}_{2}$ with the following properties:

    1. (a)

${\mathbb{M}}_{1}\leq_{\aleph_{0}}{\mathbb{M}}_{2}$,

    2. (b)

$\|{\mathbb{M}}_{2}\|\leq\|{\mathbb{M}}_{1}\|+\|{\mathfrak{m}}\|+|{{\mathfrak{E}}}^{\prime}|$,

    3. (c)

if $x\in{\mathbb{M}}_{1}$, $n<\omega$ and
${\mathfrak{e}}\in{{\mathfrak{E}}}^{\prime}$ then

${\mathbb{M}}_{2}\models\varphi^{{\mathfrak{e}},\infty}_{n}(x)\iff{\mathbb{M}}_{1}\models\varphi^{\mathfrak{e}}_{n}[x],$

    4. (d)

${\mathbb{M}}_{2}$ is the free sum of
$\\{{\mathbb{M}}_{2,i}:i<i^{\ast}\\}\cup\\{{\mathbb{M}}_{1}\\}$ where each
${\mathbb{M}}_{2,i}$ isomorphic to some member of ${\mathcal{K}}$,

    5. (e)

For each ${\mathbb{N}}\in\mathcal{K},$ there are
$||{\mathbb{M}}_{2}||$-bimodules from $\\{{\mathbb{M}}_{2,i}:i<i^{\ast}\\}$
each of them being isomorphic to ${\mathbb{N}}$.

###### Proof.

Clause (1) can be proved easily. To prove (2), let
$\kappa:=|{\mathbb{M}}_{1}\|+\|{\mathfrak{m}}\|+|{{\mathfrak{E}}}^{\prime}|$
and set

${\mathbb{M}}_{2}:={\mathbb{M}}_{1}\oplus\bigoplus_{{\mathbb{N}}\in\mathcal{K}}\bigoplus_{i<\kappa}{\mathbb{M}}_{i}^{{\mathbb{N}}},$

where each ${\mathbb{M}}_{i}^{{\mathbb{N}}}$ is isomorphic to ${\mathbb{N}}.$
It is easily seen that ${\mathbb{M}}_{2}$ is as required. ∎

We now introduce the notion of (semi) nice construction. This concept plays an
important role for the proof of our main results.

######  Definition 3.32.

Suppose ${\mathfrak{m}}=(\mathcal{K},{\mathbb{M}}_{*},{\mathfrak{E}},{\bf
R},{\bf S},{\bf T})$ is a context,
$\lambda\geq\kappa=\mathop{\mathrm{cf}}(\kappa)>\kappa({\mathfrak{E}})$. Also,
$\lambda=\lambda^{\kappa}$ and for all $\alpha<\lambda$,
$|\alpha|^{\aleph_{0}}<\lambda$. Here,
$\bar{\gamma}^{*}=\langle\gamma^{*}_{\alpha}:\alpha\leq\kappa\rangle$ is
increasing continuous and $S\subseteq S^{\kappa}_{\aleph_{0}}$777where
$S^{\kappa}_{\aleph_{0}}=\\{\alpha<\kappa:\mathop{\mathrm{cf}}(\alpha)=\aleph_{0}\\}$.
is stationary such that $S^{\kappa}_{\aleph_{0}}\setminus S$ is stationary as
well. Let also $\langle\mathfrak{e}_{\alpha}:\alpha<|{\mathfrak{E}}|\rangle$
be a fixed enumeration of ${\mathfrak{E}}$ and
$\emph{rep}(\mathcal{K})=\langle{\mathbb{N}}^{\beta}:\beta<|\mathcal{K}/\cong|\rangle$
be an enumeration of $\mathcal{K}$ up to isomorphism.

A _weakly semi-nice construction_ $\mathcal{A}$ for
$(\lambda,{{\mathfrak{m}}},S,\bar{\gamma}^{*})$ is a sequence
$\bar{\mathbb{M}}=\langle{\mathbb{M}}_{\alpha}:\alpha\leq\kappa\rangle$
together with the other objects mentioned below satisfying the following
conditions

  1. (A)

${\mathbb{M}}_{\alpha}$ is a bimodule whose universe is the ordinal
$\gamma_{\alpha}^{*}$ for $\alpha\leq\kappa$.

  2. (B)

$\alpha<\beta\ \Rightarrow\
{\mathbb{M}}_{\alpha}\subseteq{\mathbb{M}}_{\beta}$.

  3. (C)

if $\delta\leq\kappa$ is a limit ordinal, then
${\mathbb{M}}_{\delta}=\bigcup\limits_{\alpha<\delta}{\mathbb{M}}_{\alpha}$.

  4. (D)

${\mathbb{M}}_{0}={\mathbb{M}}_{\ast}$.

  5. (E)

for every $\alpha\in\kappa\setminus S$,
${\mathbb{M}}_{\alpha+1}={\mathbb{M}}_{\alpha}\oplus\mathop{\bigoplus}\limits_{t\in
J_{\alpha}}{\mathbb{N}}^{\alpha}_{t}$ where

     * (a)

$|J_{\alpha}|\leq\lambda$,

     * (b)

$J_{\alpha}\cap(\bigcup\limits_{\beta<\alpha}J_{\beta})=\emptyset,$

     * (c)

$\mathop{\mathrm{cf}}(\gamma^{*}_{\alpha+1})<\lambda\Leftrightarrow|J_{\alpha}|<\lambda$,

     * (d)

for every $t\in J_{\alpha}$ there is $\beta<|\mathcal{K}/\cong|$ such that
${\mathbb{N}}_{t}^{\alpha}\cong{\mathbb{N}}^{\beta}$, i.e., every
${\mathbb{N}}_{t}^{\alpha}$ is isomorphic to some member of $\mathcal{K},$

     * (e)

for every $\beta<|\mathcal{K}/\cong|$ there are
$||{\mathbb{M}}_{\alpha}||$-many $t\in J_{\alpha}$ such that
${\mathbb{N}}_{t}^{\alpha}\cong{\mathbb{N}}^{\beta}$.

By the assumption (b), for each
$t\in\bigcup\limits_{\beta\leq\alpha}J_{\beta}$ there is a unique
$\beta\leq\alpha$ such that $t\in J_{\beta}$; so we may replace
${\mathbb{N}}^{\beta}_{t}$ by ${\mathbb{N}}_{t}.$ Also given $t\in
J_{\alpha}$, there is a unique ${\mathbb{N}}_{t}^{\rm
so}\in\emph{rep}(\mathcal{K})$ such that ${\mathbb{N}}_{t}^{\rm
so}\cong{\mathbb{N}}_{t}^{\alpha}$. Also, we use the notations $h_{t}$ and
$h_{\alpha,t}:{\mathbb{N}}_{t}^{\rm
so}\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}{\mathbb{N}}_{t}^{\alpha}$
for the mentioned isomorphism.

  6. (F)

for $\delta\in S$, either the demand in clause888 we can ignore this first
possibility as we can just shrink $S$ if the result is stationary. (E) holds
or else there are $\gamma^{**}_{\delta}<\delta$ and

$\langle({\mathfrak{e}}_{s},{\mathbb{P}}_{s},\bar{\alpha}_{s},\bar{t}_{s},\bar{g}_{s},h_{s},q_{s}):s\in
J_{\delta}\rangle=\langle{\mathfrak{e}}^{\delta}_{s},{\mathbb{P}}^{\delta}_{s},\bar{\alpha}^{\delta}_{s},\bar{t}^{\delta}_{s},\bar{g}^{\delta}_{s},h^{\delta}_{s},q^{\delta}_{s}):s\in
J_{\delta}\rangle$

such that $J_{\delta}\cap(\bigcup_{\alpha<\delta}J_{\alpha})=\emptyset$ and

    1. (a)

${\mathfrak{e}}_{s}\in{\mathfrak{E}}$ and in fact
${\mathfrak{e}}_{s}\in\\{{\mathfrak{e}}_{\beta}:\beta<\gamma^{**}_{\delta}$
and $\beta<|{\mathfrak{E}}|\\}$,

    2. (b)

$\bar{\alpha}_{s}=\langle\alpha_{s,n}:n<\omega\rangle$,

    3. (c)

$\bar{t}=\langle t_{s,n}:n<\omega\rangle$,

    4. (d)

$\langle\alpha_{s,n}:n<\omega\rangle$ is an increasing sequence of ordinals
bigger than $\gamma^{**}_{\delta}$ such that $\alpha_{s,n}\notin S$,

    5. (e)

$\delta=\mathop{\mathrm{sup}}\\{\alpha_{s,n}:n<\omega\\}$,

    6. (f)

$t_{s,n}\in J_{\alpha_{s,n}}$,

    7. (g)

${\mathbb{N}}^{so}_{t_{s,n}}={\mathbb{N}}^{{\mathfrak{e}}_{s}}_{n}$,

    8. (h)

if $s_{1}\not=s_{2}$ are in $J_{\delta}$ then the sets

$\\{t_{s_{1},n}:n<\omega\\},\\{t_{s_{2},n}:n<\omega\\}$

are tree-like, i.e.,

       * •

$\\{t_{s_{1},n}:n<\omega\\}\cap\\{t_{s_{2},n}:n<\omega\\}$ is finite,

       * •

if $t_{s_{1},n_{1}}=t_{s_{2},n_{2}}$ then $n_{1}=n_{2}$ and
$\bigwedge\limits_{n<n_{1}}t_{s_{1},n}=t_{s_{2},n}$,

    9. (i)

$\bar{g}_{s}=\langle g_{s,n}:n<\omega\rangle$, where each $g_{s,n}$ is a
(bimodule) homomorphism from ${\mathbb{N}}^{{\mathfrak{e}}_{s}}_{n}$ into
${\mathbb{M}}_{\gamma^{**}_{\delta}}$,

    10. (j)

for $s\in J_{\delta}$, $h_{s}$ is a bimodule homomorphism from
${\mathbb{P}}^{{\mathfrak{e}}_{s}}$999where
${\mathbb{P}}^{{\mathfrak{e}}_{s}}$ is as in Lemma 3.28. onto
${\mathbb{P}}^{\delta}_{s}$, where ${\mathbb{P}}^{\delta}_{s}$ is a sub-
bimodule of ${\mathbb{M}}_{\delta+1}$,

    11. (k)

the following diagram commutes for $n<\omega$, $s\in J_{\delta}$ and
${\mathfrak{e}}={\mathfrak{e}}_{s}$ (where ${\mathbb{P}}^{\mathfrak{e}},{\bf
h}^{\mathfrak{e}}_{n}$ are from Lemma 3.28 and the $h_{t_{s,n}}$ are from
clause (E)):

$(h_{t_{s,n}}+g^{\delta}_{s,n})$${\mathbb{N}}^{\mathfrak{e}}_{n}$${\mathbb{M}}_{\delta}\cap{\mathbb{P}}^{\delta}_{s}$id${\mathbb{P}}^{\delta}_{s}$${\mathbb{P}}^{\mathfrak{e}}$${\bf
h}^{\mathfrak{e}}_{n}$$h^{\delta}_{s}$

    12. (l)

for $s\in J_{\delta}$ we have

       * (a)

${\mathbb{P}}^{\delta}_{s}\subseteq{\mathbb{M}}_{\delta+1}$,

       * (b)

${\mathbb{M}}_{\delta+1}$ is generated by
${\mathbb{M}}_{\delta}\cup\bigcup\limits_{s\in
J_{\delta}}{\mathbb{P}}^{\delta}_{s}$,

       * (c)

${\mathbb{P}}^{\delta}_{s}$,
$\langle{\mathbb{M}}_{\delta}\cup\bigcup\limits_{s^{\prime}\neq
s}{\mathbb{P}}^{\delta}_{s^{\prime}}\rangle_{{\mathbb{M}}_{\delta+1}}$ are
amalgamated freely over

$\sum\limits_{n<\omega}\mathop{\mathrm{Rang}}(h^{\delta}_{s}\circ{\bf
h}^{{\mathfrak{e}}_{s}}_{n}),$

    13. (m)

for $s\in J_{\delta}$ the type $q_{s}$ with the free variable $y$ has the form
$\\{\varphi^{\mathfrak{e}}_{n}(y-z_{s,n}):n<\omega\\}$ where
${\mathfrak{e}}\in{\mathfrak{E}}$ and $z_{s,n}\in{\mathbb{M}}_{\delta}$,

    14. (n)

$q_{s}$ is omitted by ${\mathbb{M}}_{\alpha}$ for $\alpha\in[\delta,\kappa]$,
i.e. there is no $y\in{\mathbb{M}}_{\alpha}$ such that for all
$n<\omega,~{}{\mathbb{M}}_{\alpha}\models\varphi^{\mathfrak{e}}_{n}(y-z_{s,n}).$

In abuse of notation we sometimes use
$\langle{\mathbb{M}}_{\alpha}:\alpha\leq\kappa\rangle$ as being
${\mathcal{A}}$, we of course may write
${\mathbb{M}}_{\alpha}^{\mathcal{A}},J^{\mathcal{A}}_{\alpha}$ etc. and even
$\kappa=\kappa^{\mathcal{A}},\gamma_{\alpha}=\gamma_{\alpha}^{\mathcal{A}},\lambda=\lambda^{\mathcal{A}}$.
We now define various refinements of the above concept:

######  Definition 3.33.

Using the notation of Definition 3.32,

  1. (1)

    1. (a)

By a weakly semi-nice construction for $(\lambda,{\mathfrak{m}},S,\kappa)$ we
mean a weakly semi-nice construction for
$(\lambda,{\mathfrak{m}},S,\bar{\gamma}^{*})$ where one of the following
occurs:

$\bullet$
$\kappa=\mathop{\mathrm{cf}}(\lambda),~{}(\forall\alpha<\kappa)(\gamma^{*}_{\alpha}<\lambda)$
and if $\alpha<\beta<\kappa$ then
$\gamma^{*}_{\alpha+1}-\gamma^{*}_{\alpha}\leq\gamma^{*}_{\beta+1}-\gamma^{*}_{\beta}$,

$\bullet$ $\kappa\not=\mathop{\mathrm{cf}}(\lambda)$ and
$(\forall\alpha<\lambda)$ we have
$\gamma^{*}_{\alpha}+\lambda\leq\gamma^{*}_{\alpha+1}<(\lambda^{\aleph_{0}})^{+}$.

    2. (b)

if we omit $\kappa$, i.e., write $(\lambda,{\mathfrak{m}},S)$ we mean
$\kappa=\mathop{\mathrm{cf}}(\lambda)$.

  2. (2)

We omit “semi” from “weakly semi-nice construction ${\mathcal{A}}$” if
whenever $\delta\in S$ and $s_{1}$, $s_{2}\in J_{\delta}$ then
$\bar{g}^{\delta}_{s_{1}}=\bar{g}^{\delta}_{s_{2}}$ and
${\mathfrak{e}}_{s_{1}}={\mathfrak{e}}_{s_{2}}$.

  3. (3)

We omit “weakly” from “weakly semi-nice construction ${\mathcal{A}}$” if we
add

    1. $(G)_{1}$

if ${\bf f}$ is an endomorphism of ${\mathbb{M}}_{\kappa}$ as an ${\bf
R}$-module, ${\mathfrak{e}}\in{\mathfrak{E}}^{\mathfrak{m}},\gamma<\kappa$ and
${\bf g_{n}}$ is a (bimodule) homomorphism from
${\mathbb{N}}^{\mathfrak{e}}_{n}$ into ${\mathbb{M}}_{\gamma}$ for each
$n=1,2,\ldots$, then there are $y\in{\mathbb{M}}_{\kappa}$,
$z\in{\mathbb{P}}^{\mathfrak{e}}$ and $\\{z_{n,i,\ell}:n,\ell<\omega,i<2\\}$
with $z_{n,i,\ell}\in{\rm Dom\/}({\bf
g_{\ell}})={\mathbb{N}}^{\mathfrak{e}}_{\ell}$ such that for each $n$ and all
large enough $\ell,z_{n,i,\ell}=0$ and for all large enough $n$,

      1. (a)

$z\in\sum\limits_{\ell<\omega}{\bf
h}_{\ell}^{\mathfrak{e}}(z_{n,1,\ell}-z_{n,0,\ell})+\varphi_{n}^{\mathfrak{e}}({\mathbb{P}}^{\mathfrak{e}}).$

      2. (b)

$\sum\limits_{\ell<n}{\bf f}({\bf g}_{\ell}(x_{\ell}^{\mathfrak{e}}))\in
y+\sum\limits_{\ell<\omega}{\bf
g}_{\ell}(z_{n,1,\ell})+\varphi_{n}^{\mathfrak{e}}({\mathbb{M}}_{\kappa}).$

    2. $(G)_{2}$

if ${\mathfrak{e}}\in{\mathfrak{E}}^{\mathfrak{m}}$ and for $n<\omega$ and
$\alpha\in\kappa\smallsetminus S$, ${\bf h}_{\alpha,n}$ is an embedding of
${\mathbb{N}}^{\mathfrak{e}}_{n}$ into ${\mathbb{M}}_{\kappa}$ such that
${\mathbb{M}}_{\alpha}+\mathop{\mathrm{Rang}}({\bf
h}_{\alpha,n})={\mathbb{M}}_{\alpha}\oplus\mathop{\mathrm{Rang}}({\bf
h}_{\alpha,n})\leq_{\aleph_{0}}{\mathbb{M}}_{\kappa}$, then there are an
embedding ${\bf h}$ of ${\mathbb{P}}^{\mathfrak{e}}$ into
${\mathbb{M}}_{\kappa}$ and ordinals $\alpha_{n}\in\kappa\smallsetminus S$ for
$n=1,2,\ldots$ such that

${\bf h}_{\alpha_{n},n}={\bf h}\circ{\bf h}^{\mathfrak{e}}_{n}$

for each such $n$.

  4. (4)

$\bar{\mathbb{M}}=\langle{\mathbb{M}}_{\alpha}:\alpha\leq\kappa\rangle$ is
strongly semi-nice construction for $(\lambda,{\mathfrak{m}},S,\kappa)$ if
${\bf f}$ is an ${\bf R}$-endomorphism of ${\mathbb{M}}_{\kappa}$ and
$\mathfrak{e}\in{\mathfrak{E}}^{{{\mathfrak{m}}}},$ then there are
$n(\ast)<\omega,\alpha<\lambda$ and
$z\in{\mathbb{N}}_{n(\ast)}^{\mathfrak{e}}$ such that for every bimodule
homomorphism
$h:{\mathbb{N}}_{n(\ast)}^{\mathfrak{e}}\to{\mathbb{M}}_{\kappa},$

${\bf
f}(h(x_{n}^{\mathfrak{e}}))-h(z)\in{\mathbb{M}}_{\alpha}+\varphi_{\omega}^{\mathfrak{e}}({\mathbb{M}}_{\kappa}).$

  5. (5)

We say ${\mathbb{M}}$ is weakly semi-nice (resp. semi-nice) for
$(\lambda,S,{\mathfrak{m}},\kappa)$ if for some weakly semi-nice (resp. semi-
nice) construction
$\bar{\mathbb{M}}=\langle{\mathbb{M}}_{\alpha}:\alpha\leq\kappa\rangle$ for
$(\lambda,S,{\mathfrak{m}},\kappa)$ we have
${\mathbb{M}}={\mathbb{M}}_{\kappa}$. If we omit $S$ we mean for some
$S\subseteq S^{\kappa}_{\aleph_{0}}$, similarly for $\lambda$ and $\kappa.$

######  Lemma 3.34.

Let $\langle{\mathbb{M}}_{\alpha}:\alpha\leq\kappa\rangle$ be a semi-nice
construction for $(\lambda,{{\mathfrak{m}}},S,\bar{\gamma}^{*})$ and
$\kappa({\mathfrak{E}}^{{{\mathfrak{m}}}})={\aleph_{0}}$, (not used in part
(1)).

  1. (1)

if $\alpha\notin S$ and $\alpha\leq\beta\leq\kappa$, then
${\mathbb{M}}_{\alpha}\leq_{\aleph_{0}}{\mathbb{M}}_{\beta}$, i.e.,
${\mathbb{M}}_{\alpha}$ is an almost direct ${\mathcal{K}}$-summand of
${\mathbb{M}}$ with respect to $\aleph_{0}$.

  2. (2)

If $\alpha\leq\beta\leq\kappa$ and $n<\omega$ then
$\varphi^{\mathfrak{e}}_{n}({\mathbb{M}}_{\alpha})={\mathbb{M}}_{\alpha}\cap\varphi^{\mathfrak{e}}_{n}({\mathbb{M}}_{\beta})$.

  3. (3)

Assume ${\mathfrak{e}}\in{\mathfrak{E}}^{{{\mathfrak{m}}}}$, $\delta\in S$ and
clause (F) of Definition 3.32 holds for $\delta$. Suppose
$s_{0},\ldots,s_{k(*)-1}$ are distinct members of $J_{\delta}$. Then

    1. (a)

for each $n$, the set
$\varphi^{\mathfrak{e}}_{n}(\langle{\mathbb{M}}_{\delta}\cup\bigcup\\{{\mathbb{P}}_{s_{k}}^{\mathfrak{e}}:k<k(*)i\\}\rangle_{{\mathbb{M}}_{\delta+1}})$
is equal to

$\langle{\mathbb{M}}_{\delta}\cup\bigcup\\{{\mathbb{P}}^{\mathfrak{e}}_{s_{k}}:k<k(*)\\}\rangle_{{\mathbb{M}}_{\delta+1}})\cap\varphi^{\mathfrak{e}}_{n}({\mathbb{M}}_{\kappa})\rangle.$

    2. (b)

Suppose $z_{\ell}\in{\mathbb{P}}^{\mathfrak{e}}$ for
$\ell<n,z\in{\mathbb{M}}_{\delta}$, and
$z+\sum\\{h^{\delta}_{s_{k}}(z_{k}):k<k(*)\\}\in\varphi_{n}({\mathbb{M}}_{\delta+1})$.
Then, there is $z^{\prime}_{k}\in\sum\\{{\rm
Rang}(h^{\mathfrak{e}}_{\ell}):\ell<\omega\\}$ such that the following two
properties hold:

      1. $(b.1)$

: $z_{k}-z^{\prime}_{k}\in\varphi_{n}({\mathbb{P}}^{\mathfrak{e}})$ and

      2. $(b.2)$

:
$z+\sum\\{h^{\delta}_{s_{k}}(z^{\prime}_{k}):k<k(*)\\}\in\varphi_{n}({\mathbb{M}}_{\delta+1})$.

    3. (c)

$\langle{\mathbb{M}}_{\delta}\cup\\{{\mathbb{P}}^{\mathfrak{e}}_{s_{k}}:k<k(*)\\}\rangle_{{\mathbb{M}}_{\delta+1}}\leq^{\rm
pr}_{\varphi_{n}^{\mathfrak{e}}}{\mathbb{M}}_{\delta+1}$.

    4. (d)

$\varphi^{\mathfrak{e}}_{n}({\mathbb{M}}_{\delta+1})=\varphi^{\mathfrak{e}}_{n}({\mathbb{M}}_{\delta})+\sum\\{\varphi^{\mathfrak{e}}_{n}({\mathbb{P}}^{\mathfrak{e}}_{s}):s\in
J_{\delta}\\}$.

###### Proof.

(1). We proceed by induction on $\beta\geq\alpha$. The straightforward details
leave to reader.

(2). If $\alpha\notin S,$ then the conclusion follows from (1) and Lemma 3.18.
Now suppose that $\alpha\in S.$ By Definition 3.32(F), the result holds if
$\beta=\alpha+1.$ But then for any $\beta>\alpha$ (and since $\alpha+1\notin
S$), we have

${\mathbb{M}}_{\alpha}\cap\varphi^{\mathfrak{e}}_{n}({\mathbb{M}}_{\beta})={\mathbb{M}}_{\alpha}\cap(\varphi^{\mathfrak{e}}_{n}({\mathbb{M}}_{\beta})\cap{\mathbb{M}}_{\alpha+1})=\varphi^{\mathfrak{e}}_{n}({\mathbb{M}}_{\alpha+1})\cap{\mathbb{M}}_{\alpha}=\varphi^{\mathfrak{e}}_{n}({\mathbb{M}}_{\alpha}).$

(3). Items (a), (c) and (d) can be proved easily by Definition 3.32(F). Let us
prove (b). To this end, take $z_{\ell}\in{\mathbb{P}}^{\mathfrak{e}}$ for
$\ell<k(*),z\in{\mathbb{M}}_{\delta}$, and
$z+\sum\\{h^{\delta}_{s_{k}}(z_{k}):k<k(*)\\}\in\varphi_{n}({\mathbb{M}}_{\delta+1})$.

In the light of (c) we observe that

$z+\sum\\{h^{\delta}_{s_{k}}(z_{k}):k<k(*)\\}\in\varphi^{\mathfrak{e}}_{n}(\langle{\mathbb{M}}_{\delta}\cup\\{{\mathbb{P}}^{\mathfrak{e}}_{s_{k}}:k<k(*)\\}\rangle_{{\mathbb{M}}_{\delta+1}}).$

Let $k<k(*)$. We now apply the items $(a)$, $(c)$ and $(d)$ to find
$y\in\varphi^{\mathfrak{e}}_{n}({\mathbb{M}}_{\delta})$ and
$y_{k}\in\varphi^{\mathfrak{e}}_{n}({\mathbb{P}}^{\mathfrak{e}}_{s_{k}})$ such
that

$z+\sum\\{h^{\delta}_{s_{k}}(z_{k}):k<k(*)\\}=y+\sum\\{y_{k}:k<k(*)\\}.$

Since $\varphi^{\mathfrak{e}}_{n}({\mathbb{P}}^{\mathfrak{e}}_{s_{k}})$ is the
image of $\varphi^{\mathfrak{e}}_{n}({\mathbb{P}}^{\mathfrak{e}})$ under
$h_{s_{k}}^{\delta}$, we have $y_{k}=h_{s_{k}}^{\delta}(x_{k})$ for some
$x_{k}\in\varphi^{\mathfrak{e}}_{n}({\mathbb{P}}^{\mathfrak{e}}).$ Then

$\sum\\{h^{\delta}_{s_{k}}(z_{k}-x_{k}):k<k(*)\\}=y-z\in\varphi^{\mathfrak{e}}_{n}({\mathbb{M}}_{\delta}).$

As $s_{0},\cdots,s_{k(*)-1}$ are chosen distinct, we deduce that

$h^{\delta}_{s_{k}}(z_{k}-x_{k})\in\mathop{\mathrm{Rang}}(h^{\delta}_{s_{k}})\cap\varphi^{\mathfrak{e}}_{n}({\mathbb{M}}_{\delta})=\varphi^{\mathfrak{e}}_{n}(\mathop{\mathrm{Rang}}(h^{\delta}_{s_{k}})),$

where $k<k(*).$ Hence,
$z_{k}-x_{k}\in\varphi^{\mathfrak{e}}_{n}({\mathbb{P}}^{\mathfrak{e}})$.

Let $h<\omega$ be such that for each $k<k(*)$,

$\begin{array}[]{ll}x_{k}&\in\varphi^{\mathfrak{e}}_{n}\left(\bigoplus\limits_{\ell\leq
h}(\mathop{\mathrm{Rang}}({\bf f}^{\mathfrak{e}}_{n}))\right)\\\
&=\bigoplus\limits_{\ell\leq
h}\varphi^{\mathfrak{e}}_{n}(\mathop{\mathrm{Rang}}({\bf
f}^{\mathfrak{e}}_{\ell}))\\\
&=\bigoplus\limits_{\ell<h}\varphi^{\mathfrak{e}}_{n}(\mathop{\mathrm{Rang}}({\bf
h}^{\mathfrak{e}}_{\ell}))\oplus\varphi^{\mathfrak{e}}_{n}(\mathop{\mathrm{Rang}}({\bf
f}^{\mathfrak{e}}_{h})).\end{array}$

Thus, for some
$z^{\prime}_{k}\in\bigoplus\limits_{\ell<h}\varphi^{\mathfrak{e}}_{n}(\mathop{\mathrm{Rang}}({\bf
h}^{\mathfrak{e}}_{\ell}))$, we have

$x_{k}-z^{\prime}_{k}\in\varphi^{\mathfrak{e}}_{n}(\mathop{\mathrm{Rang}}({\bf
f}^{\mathfrak{e}}_{h}))\subseteq\varphi^{\mathfrak{e}}_{n}({\mathbb{P}}^{\mathfrak{e}}).$

It then follows that

$z_{k}-z^{\prime}_{k}=(z_{k}-x_{k})+(x_{k}-z^{\prime}_{k})\in\varphi^{\mathfrak{e}}_{n}({\mathbb{P}}^{\mathfrak{e}})+\varphi^{\mathfrak{e}}_{n}({\mathbb{P}}^{\mathfrak{e}})=\varphi^{\mathfrak{e}}_{n}({\mathbb{P}}^{\mathfrak{e}}).$

We now show that
$w:=z+\sum\\{h^{\delta}_{s_{k}}(z^{\prime}_{k}):k<k(*)\\}\in\varphi_{n}({\mathbb{M}}_{\delta+1}).$
Indeed,

$\begin{array}[]{ll}w&=z+\sum\\{h^{\delta}_{s_{k}}(z^{\prime}_{k}-z_{k}):k<k(*)\\}+\sum\\{h^{\delta}_{s_{k}}(z_{k}):k<k(*)\\}\\\
&=(z+\sum\\{h^{\delta}_{s_{k}}(z_{k}):k<k(*)\\})+\sum\\{h^{\delta}_{s_{k}}(z^{\prime}_{k}-z_{k}):k<k(*)\\}\\\
&\in\varphi_{n}({\mathbb{M}}_{\delta+1})+\sum\\{\varphi^{\mathfrak{e}}_{n}({\mathbb{P}}^{\delta}_{s_{k}}):k<k(*)\\}\\\
&=\varphi_{n}({\mathbb{M}}_{\delta+1}).\end{array}$

This completes the proof of (b) and hence of the lemma. ∎

The main result of this section is that under suitable assumptions on
$(\lambda,{{\mathfrak{m}}},S,\bar{\gamma}^{*})$, there is, in $\text{ZFC},$ a
semi-nice construction for it. Before, we state and prove our result, let us
show that under extra set theoretic assumptions we can get a stronger result.

######  Definition 3.35.

Suppose $\lambda=\mathop{\mathrm{cf}}(\lambda)>|{\bf R}|+|{\bf
S}|+\aleph_{0}$, $S\subseteq S^{\lambda}_{\aleph_{0}}$ is stationary and non-
reflecting101010$S$ is non-reflecting if for all limit ordinals $\xi<\lambda$
of uncountable cofinality, $S\cap\xi$ is non-stationary. such that
$S^{\lambda}_{\aleph_{0}}\setminus S$ is stationary as well. Then
$\bar{\mathbb{M}}=\langle{\mathbb{M}}_{\alpha}:\alpha\leq\lambda\rangle$ is
called very nicely constructed for $(\lambda,{\mathfrak{m}},S)$ if:

  1. (1)

Clauses (A)-(E) of Definition 3.32 holds,

  2. (2)

Each $J_{\alpha}$ is singleton,

  3. (3)

For each ${\mathbb{N}}\in{\mathcal{K}}$, for stationary many
$\alpha\in\lambda\setminus S$, ${\mathbb{N}}$ appears as one of the summands
of ${\mathbb{M}}_{\alpha+1}$,

  4. (4)

Replace clause (F) of Definition 3.32 by the following modification:
For $\delta\in S$, ${\mathbb{M}}_{\delta+1}$ is defined either as in (E), or
else there are an infinite $\mathcal{U}\subseteq\omega$ and the sequences
$\langle\alpha_{n}:n<\omega\rangle$ and
$\langle\beta_{m}(\alpha_{n}):m\in\mathcal{U}\rangle$ and $\langle
h_{\alpha_{n},n}:n\in\mathcal{U}\rangle$ such that

    1. (a)

each $\alpha_{n}\in\lambda\setminus S$,

    2. (b)

$\langle\alpha_{n}:n<\omega\rangle$ is increasing and cofinal in $\delta,$

    3. (c)

${\mathfrak{e}}_{\delta}={\mathfrak{e}}$,

    4. (d)

for each $n<\omega$ and $m\in\mathcal{U}$,
$\alpha_{n}<\beta_{m}(\alpha_{n})<\alpha_{n+1}$ is in $\lambda\setminus S$,

    5. (e)

for $n\in\mathcal{U}$,
$h_{\alpha_{n},n}:{\mathbb{N}}_{n}^{\mathfrak{e}}\to{\mathbb{M}}_{\beta_{n}(\alpha_{n})}$
is a bimodule homomorphism and
${\mathbb{M}}_{\alpha_{n}}+\mathop{\mathrm{Rang}}(h_{\alpha_{n},n})={\mathbb{M}}_{\alpha_{n}}\oplus\mathop{\mathrm{Rang}}(h_{\alpha_{n},n})\leq_{\aleph_{0}}{\mathbb{M}}_{\beta_{m}(\alpha_{n})}$.

    6. (f)

Set
${\mathbb{P}}_{\mathcal{U}}:=\bigoplus_{n\in\mathcal{U}}\mathop{\mathrm{Rang}}({\bf
f}_{n}^{\mathfrak{e}})$ and
${\mathbb{P}}_{{\mathcal{U}},n}:=\sum\limits_{\ell\in\mathcal{U}\cap
n}\mathop{\mathrm{Rang}}({\bf h}^{\mathfrak{e}}_{\ell})$. Recall from Lemma
3.28 the bimodule homomorphisms ${\bf
h}^{\mathfrak{e}}_{n}:{\mathbb{N}}^{\mathfrak{e}}_{n}\longrightarrow{\mathbb{P}}^{\mathfrak{e}}$
and ${\bf
f}^{\mathfrak{e}}:{\mathbb{N}}^{\mathfrak{e}}_{0}\longrightarrow{\mathbb{P}}^{\mathfrak{e}}$.
We define
${\mathbb{N}}^{\ast}_{\delta}:=\sum\limits_{n\in{\mathcal{U}}}h_{\alpha_{n},n}({\mathbb{N}}_{n})$.
Then ${\mathbb{P}}^{\mathfrak{e}}$ is isomorphic to
${\mathbb{P}}_{\mathcal{U}}$ by an isomorphism $h_{\delta}$ such that the
following diagram commutes (here, $n=i(m)=m$-th member of $\mathcal{U}$):

    7. (g)

${\mathbb{P}}^{\mathfrak{e}}_{n}=h_{\delta}^{\prime\prime}({\mathbb{P}}_{{\mathcal{U}},n})$.
So in ${\mathbb{M}}_{\delta+1}$,
${\mathbb{P}}^{\mathfrak{e}}\cap{\mathbb{M}}_{\delta}={\mathbb{N}}^{\ast}_{\delta}$.

Recall that for a stationary set $S\subseteq\lambda,$ Jensen’s diamond
$\diamondsuit_{\lambda}(S)$ asserts the existence of a sequence $\langle
S_{\alpha}:\alpha\in S\rangle$ such that for every $X\subseteq\lambda$ the set
$\\{\alpha\in S:X\cap\alpha=S_{\alpha}\\}$ is stationary. It is easily seen
that $\diamondsuit_{\lambda}(S)$ implies $2^{<\lambda}=\lambda$. One the one
hand, due to a well-known theorem of Jensen, $\diamondsuit_{\lambda}(S)$ holds
in Gödel’s constructible universe L for all uncountable regular cardinals
$\lambda$ and all stationary sets $S\subseteq\lambda$. On the other hand, if
$2^{<\lambda}$ is forced to be above $\lambda,$ then
$\diamondsuit_{\lambda}(S)$ fails for all stationary sets $S\subseteq\lambda$.
The next theorem of Shelah shows that under diamond principle, we can get very
nicely $\langle{\mathbb{M}}_{\alpha}:\alpha\leq\lambda\rangle$.

######  Lemma 3.36.

Suppose ${\mathfrak{m}}$ is a non-trivial context,
$\lambda=\mathop{\mathrm{cf}}(\lambda)>|{\bf R}|+|{\bf
S}|+\kappa({\mathfrak{m}})+||{\mathfrak{m}}||$, $S\subseteq
S^{\lambda}_{\aleph_{0}}$ is a stationary non-reflecting subset of $\lambda$
and $\diamondsuit_{\lambda}(S)$ holds. Then there is a very nicely
construction for $(\lambda,{\mathfrak{m}},S)$ such that

  1. (1)

$(\lambda,{\mathfrak{m}},S)$ is strongly very nice.

  2. (2)

If $\mathop{\mathrm{cf}}(\alpha)\in\kappa\smallsetminus S$ and
$\alpha<\beta\leq\lambda$, then ${\mathbb{M}}_{\alpha}$ is a ${\mathcal{K}}$\-
directed summand of ${\mathbb{M}}_{\beta}$.

###### Proof.

See [46, Section 2]. ∎

We would like to prove a similar result in ZFC, thus we have to avoid the use
of diamond in the above lemma. To this end, we use Shelah’s Black Box. This
leads us to get a weaker conclusion than Lemma 3.36, but as we will see later,
it is sufficient to get our desired results.

######  Notation 3.37.

For the rest of this section we use the following:

  * •

$\lambda>\kappa$ be infinite cardinals.

  * •

$\mathcal{H}_{<\theta}(X)$ be the least set $Y$ such that $Y\supseteq X$ and
if $y\subseteq Y$ and $|y|<\theta$,then $y\in Y.$

  * •

$\langle\tau_{n}:n<\omega\rangle$ be an increasing sequence of vocabularies,
each of size $\leq\kappa$ such that for each $n<\omega$, there exists some
unary predicate $P_{n}$ in $\tau_{n+1}\setminus\tau_{n}$.

######  Definition 3.38.

For $n<\omega$ let $\mathcal{F}_{n}$ be the family of sets of the form

$\\{(\mathcal{N}_{\ell},f_{\ell}):\ell\leq n\\}$

satisfying the following conditions:

  1. (a)

$f_{\ell}:\kappa^{\leq\ell}\to\lambda^{\leq\ell}$ is a tree embedding, i.e.,

    1. (1)

for each $\eta\in\kappa^{\leq\ell},~{}\eta$ and $f_{\ell}(\eta)$ have the same
length,

    2. (2)

for $\eta\lhd\nu$ in $\kappa^{\leq\ell},~{}f_{\ell}(\eta)\lhd f_{\ell}(\nu)$,

    3. (3)

$~{}f_{\ell}$ is one-to-one,

  2. (b)

for $\ell+1\leq n,~{}f_{\ell+1}$ extends $f_{\ell},$

  3. (c)

for some $\tau^{\prime}_{\ell}\subseteq\tau_{\ell},~{}\mathcal{N}_{\ell}$ is a
$\tau^{\prime}_{\ell}$-structure of size $\leq\kappa$ and the universe of
$\mathcal{N}_{\ell}$, denoted by $N_{\ell}$, is a subset of
$\mathcal{H}_{<\kappa^{+}}(\lambda)$.

  4. (d)

$\tau^{\prime}_{\ell+1}\cap\tau_{\ell}=\tau^{\prime}_{\ell}$ and
$\mathcal{N}_{\ell+1}\restriction\tau^{\prime}_{\ell}$ extends
$\mathcal{N}_{\ell}$.

  5. (e)

if $P_{m}\in\tau^{\prime}_{m+1}$, then $P_{m}^{\mathcal{N}_{\ell}}=N_{\ell}.$

  6. (f)

if $x,y\in N_{\ell},$ then $\\{x,y\\}\in N_{\ell}$ and $\emptyset\in
N_{\ell}.$

  7. (g)

$\mathop{\mathrm{Rang}}(f_{\ell})\subseteq\mathcal{N}_{\ell}$.

######  Definition 3.39.

Let $\langle\mathcal{F}_{n}:n<\omega\rangle$ be as in Definition 3.38. Then
$\mathcal{F}_{\omega}$ is the family consisting of all pairs $(\mathcal{N},f)$
such that for some sequence
$\langle(\mathcal{N}_{\ell},f_{\ell}):\ell<\omega\rangle$, we have

  1. (a)

for each $n<\omega$, $\\{(\mathcal{N}_{\ell},f_{\ell}):\ell\leq n\\}$ belongs
to $\mathcal{F}_{n},$

  2. (b)

$f=\bigcup\limits_{\ell<\omega}f_{\ell}$ and
$\mathcal{N}=\bigcup\limits_{\ell<\omega}\mathcal{N}_{\ell}.$

We may note that if for each $m<\omega,P_{m}\in\tau^{\prime}_{m+1}$, then for
each $(\mathcal{N},f)\in\mathcal{F}_{\omega}$, there is a unique sequence
$\langle(\mathcal{N}_{\ell},f_{\ell}):\ell<\omega\rangle$ witnessing this.

######  Definition 3.40.

Let $(\mathcal{N},f)\in\mathcal{F}_{\omega}$. A branch of $f$ is any
$\eta\in\lambda^{\omega}$ such that for each $n<\omega,~{}\eta\restriction
n\in\mathop{\mathrm{Rang}}(f)$. We use $\lim(f)$ for the set of all branches
of $f$.

Given $W\subseteq\mathcal{F}_{\omega}$, we define two games $\Game_{W}$ and
$\Game^{\prime}_{W}$ as follows.

######  Definition 3.41.

Suppose $W\subseteq\mathcal{F}_{\omega}$.

  1. (1)

The game $\Game_{W}$ of length $\omega$ is defined as follows. In the $n^{\rm
th}$ move, player I chooses $\mathcal{N}_{n}$ such that
$\\{(\mathcal{N}_{\ell},f_{\ell}):\ell\leq n\\}\in\mathcal{F}_{n}$ (noting
that $f_{0}$ is determined), and player II chooses a tree embedding
$f_{n}:\kappa^{\leq n}\to\lambda^{\leq n}$ extending
$\bigcup\limits_{\ell<n}f_{\ell}$ such that
$\mathop{\mathrm{Rang}}(f_{n})\setminus(\bigcup\limits_{\ell<n}\mathop{\mathrm{Rang}}(f_{\ell}))$
is disjoint to $\bigcup\limits_{\ell<n}N_{\ell}.$ Player II wins if
$(\bigcup\limits_{\ell<\omega}\mathcal{N}_{\ell},\bigcup\limits_{\ell<\omega}f_{\ell})\in
W.$

  2. (2)

The game $\Game^{\prime}_{W}$ of length $\omega$ is defined as follows. In the
zero move, player I chooses $k<\omega$,
$\\{(\mathcal{N}_{\ell},f_{\ell}):\ell\leq k\\}\in\mathcal{F}_{k}$, and
$X_{0}\subseteq\lambda^{<\omega}$ of size less than $\lambda$. For $n>0,$ in
the $n^{\rm th}$ move, player I chooses $\mathcal{N}_{k+n}$ and $X_{n}$ such
that $\\{(\mathcal{N}_{\ell},f_{\ell}):\ell\leq k+n\\}\in\mathcal{F}_{k+n}$
and $X_{n}\subseteq\lambda^{<\omega}$ is of size less than $\lambda$. Then
player II chooses a tree embedding $f_{k+n}:\kappa^{\leq k+n}\to\lambda^{\leq
k+n}$ extending $\bigcup\limits_{\ell<k+n}f_{\ell}$ such that
$\mathop{\mathrm{Rang}}(f_{k+n})\setminus(\bigcup\limits_{\ell<k+n}\mathop{\mathrm{Rang}}(f_{\ell}))$
is disjoint to
$\bigcup\limits_{\ell<n}N_{\ell}\cup\bigcup\limits_{\ell<n}X_{\ell}.$ Player
II wins if
$(\bigcup\limits_{\ell<\omega}\mathcal{N}_{\ell},\bigcup\limits_{\ell<\omega}f_{\ell})\in
W.$

######  Definition 3.42.

Suppose $W\subseteq\mathcal{F}_{\omega}$. Recall that

  1. (1)

$W$ is called a barrier if player I does not win $\Game_{W}$ or even
$\Game^{\prime}_{W}$.

  2. (2)

$W$ is called a strong barrier if player II wins $\Game_{W}$ and even
$\Game^{\prime}_{W}$.

  3. (3)

$W$ is called disjoint if for distinct
$(\mathcal{N}^{1},f^{1}),(\mathcal{N}^{2},f^{2})$ in $W$, $f^{1}$ and $f^{2}$
have no common branch.

We are now ready to state the version of Shelah’s black box theorem that is
needed in this paper.

######  Lemma 3.43 (The Black Box theorem).

Suppose $\lambda>\kappa$ are infinite cardinals,
$\mathop{\mathrm{cf}}(\lambda)>\aleph_{0},~{}\lambda^{\aleph_{0}}=\lambda^{\kappa}$
and $S\subseteq S^{\lambda}_{\aleph_{0}}$ is stationary. Then there is a
sequence

$W=\\{(\mathcal{N}^{\alpha},f^{\alpha}):\alpha<\alpha_{*}\\}\subseteq\mathcal{F}_{\omega}$

and a non-decreasing function

$\zeta:\alpha_{*}\to S$

such that the following properties are satisfied:

  1. (1)

$W$ is a disjoint strong barrier,

  2. (2)

every branch of $f^{\alpha}$ is an increasing sequence converging to
$\zeta(\alpha)\in S$,

  3. (3)

each $\mathcal{N}^{\alpha}$ is transitive,

  4. (4)