# On tight submodules
of modules over valuation domains

Peter Danchev  Institute of Mathematics & Informatics, Bulgarian Academy of
Sciences, 1113 Sofia, Bulgaria<EMAIL_ADDRESS><EMAIL_ADDRESS>and
László Fuchs  Department of Mathematics, Tulane University, New Orleans,
Louisiana 70118, USA and  1724 Pinetree Cir NE, Atlanta, Georgia 30329, USA
<EMAIL_ADDRESS>
###### Abstract.

This note offers an unusual approach of studying a class of modules inasmuch
as it is investigating a subclass of the category of modules over a valuation
domain. This class is far from being a full subcategory, it is not even a
category. Our concern is the subclass consisting of modules of projective
dimension not exceeding one, admitting only morphisms whose kernels and
cokernels are also objects in this subclass. This class is still tractable,
several features are in general simpler than in module categories, but lots of
familiar properties are lost. A number of results on modules in this class are
similar to those on modules over rank one discrete valuation domains (where
the global dimension is 1). The study is considerably simplified by taking
advantage of the general theory of modules over valuation domains available in
the literature, e.g. in [14]-[15]. Our main goal is to establish the basic
features and have a closer look at injectivity, pure-injectivity, and
cotorsionness, but we do not wish to enter into an in-depth study of these
properties.

###### 1991 Mathematics Subject Classification:

Primary 13C05, 13F99. Secondary 13G05

##  1\. Introduction

Everybody familiar with module theory over integral domains knows well that
the theory simplifies tremendously if the ring is a Dedekind domain; i.e. the
modules have projective dimension (p.d.) at most one. That the condition
‘p.d.1’ is so powerful was recognized by E. Matlis who developed interesting
properties under the hypothesis that the field of quotients as a module has
p.d.1; see [17] (in [15] these rings were named Matlis domains in his honor).
In order to understand p.d.1 better and to learn more about it, it is natural
to try to find out how the theory would look like over a general integral
domain if one deals only with its modules of p.d. not exceeding $1$, and
ignores modules of higher projective dimensions. This means to investigate a
class where the objects are modules of p.d.$\leq 1$ and morphisms are required
to have both kernels and cokernels of p.d.$\leq 1$. This is what we are
planning to do in this article over an arbitrary valuation domain (i.e. an
integral domain where the ideals form a chain with respect to inclusion).
Valuation domains are the first and obvious choice for such a study, as they
are sufficiently general, but still manageable, and luckily, there is an
extensive literature available on them that reveals a lot of information one
can take advantage of.

The selected subclass of the module category is not a category; a primary
reason for it is that the usual composition rule of mappings works only under
an additional condition. When dealing with modules of such a class, it is soon
realized that one has to reassess familiar facts and obvious concepts to fit
into the new situation. The usual sum of two morphisms is rarely another one,
submodules that belong to this subclass only exceptionally form a lattice, the
tensor product of two objects may not belong to this class, etc. But on the
other hand, some nice features of discrete rank one valuation domains carry
over, like the equality of injectivity and divisibility or the pure-
injectivity of cyclically presented torsion modules. We will also pay
attention to pure-injectivity and to the cotorsion property (Sections 7 and
8). Though these concepts are defined to have close resemblance to the
familiar module concepts, one should always keep in mind that they are not
exactly the same. The discussion of the submodules in direct sums of
cyclically presented modules (Section 5) already shows the huge difference
from the traditional treatment of modules.

Throughout the symbol $V$ will denote a valuation domain (commutative), and
$Q$ its quotient field. We will use the notation $K=Q/V$. The symbol
$V^{\times}$ denotes the multiplicative monoid of non-zero elements of $V$.
Torsion and torsion-free modules have the usual meaning. The abbreviation p.d.
denotes projective dimension. The global weak dimension of a valuation domain
is $1$. Likewise, $|X|$ denotes the cardinality of the set $X$, and ${\omega}$
is the smallest infinite ordinal. We also abbreviate gen $M$ to stand for the
minimal cardinality of generating sets of $M$. We use “countably generated” to
mean gen $M={\aleph}_{0}$ (not finite). Torsion-freeness and flatness are
equivalent over valuation domains; consequently, relative divisibility and
purity have the same meaning. For the construction of valuation domains with
prescribed value groups, see e.g. [15, Chap. II, Section 3].

For a valuation domain $V$, we consider the category ${\mathcal{C}}_{V}$ as
the category $V$-Mod of $V$-modules with the usual morphisms. Our main goal is
to study the subclass ${\mathcal{C}}_{V}^{*}$ whose objects are the
$V$-modules of p.d.$\leq 1$ and whose morphisms are required to have kernels
and cokernels that also have p.d.$\leq 1$. The subobjects are the tight
submodules (see Section 2). While our results are restricted to this subclass,
in proofs we will often use arguments and concepts from the covering category
${\mathcal{C}}_{V}$.

Modules of projective dimension one have been discussed briefly in an earlier
paper [9] with emphasis on Prüfer domains, and some of our present results
appeared there in a different context.

The idea of investigating the subclass ${\mathcal{C}}_{V}^{*}$ comes from a
recent paper [12] where tight submodules played a dominating role. To deal
with classes of modules where both the kernels and the cokernels of the maps
were restricted looked strange, but at the same time interesting and
challenging. We admit, we were first hesitating to get involved in an
uncharted territory with no immediate applications in the horizon, but in
spite of this we decided to start working on this class, since we believed
that the results could be helpful in a better understanding of the impact of
projective dimension one as well as of the role of tightness in submodules. In
addition, they might provide counterexamples in unusual situations. In this
paper we begin with exploring this idea, and accordingly, we have been trying
to find out not only what can be verified, but also what is no longer valid in
comparison to conventional module theory.

##  2\. Tight Submodules

The fundamental concept we are using throughout is the tightness of submodules
in modules of p.d.$\leq 1$. The following definition applies to all rings. Let
$B<A$ be modules such that p.d.$A\leq 1$. $B$ is called tight in $A$ if
p.d.$A/B\leq 1$. Then also p.d.$B\leq 1$. It is evident that direct summands
are tight submodules, but tight submodules need not be summands. The tightness
for p.d.1 that was introduced in [8] and immediately generalized to higher
p.d.s in [1], was slightly different: its definition required that
p.d.$A/B\leq$ p.d.$A$. The only difference between the two variants is that in
a free $V$-module cyclically presented a free submodule is not tight in the
sense of [8], but is in this paper. (The present version was called
t-submodule in [9].)

The following easily verifiable basic rules will be applied, often without
explicit reference:

######  Lemma 2.1.

Let $C<B<A$ be $V$-modules and p.d.$A\leq 1$.

(a) If $C$ is tight in $B$ and $B$ is tight in $A$, then $C$ is tight in $A$.

(b) If $C$ and $B$ are tight in $A$, then $C$ is tight in $B$.

(c) If $C$ and $B$ are tight in $A$, then $B/C$ is tight in $A/C$.

(d) If $C$ is tight in $A$ and $B/C$ is tight in $A/C$, then $B$ is tight in
$A$. ∎

As already mentioned before, the symbol ${\mathcal{C}}_{V}$ will denote the
category of $V$-modules with the usual morphisms, while
${\mathcal{C}}_{V}^{*}$ is the subclass whose objects are the $V$-modules of
p.d.$\leq 1$. E.g. the frequently used localizations $V_{S}$ at countable
multiplicative submonoids $S$ of $V^{\times}$ are objects of our class
${\mathcal{C}}_{V}^{*}$. The subobjects of an object
$M\in{\mathcal{C}}_{V}^{*}$ are the tight submodules of $M$. The morphisms in
${\mathcal{C}}_{V}^{*}$ are those module homomorphisms ${\phi}:M\to N$
($M,N\in{\mathcal{C}}_{V}^{*}$) for which Im ${\phi}$ is tight in $N$. This
also means that Ker ${\phi}$ should be a tight submodule of $M$. Thus
morphisms in ${\mathcal{C}}_{V}^{*}$ have tight kernels and images. Clearly, a
submodule $A$ of $M$ is tight if and only if the inclusion morphism $A\to M$
belongs to ${\mathcal{C}}_{V}^{*}$. In order to check whether or not
${\mathcal{C}}_{V}^{*}$ is a category, one ought to examine the axioms of
category theory; in particular, the critical one that says that if
${\alpha}:A\to B$ and ${\beta}:B\to C$ are morphisms, then so is the composite
map ${\beta}{\alpha}:A\to C$. Unfortunately, this property is seldom satisfied
in ${\mathcal{C}}_{V}^{*}$. As we will see in a moment, this property is
related to the one that the sum of two tight submodules is again tight – which
holds very rarely. In the next lemma this property is compared to similar
properties, in particular to the tightness of the intersection of two tight
submodules.

######  Lemma 2.2.

Let $M$ be a $V$-module of p.d.$\leq 1$, and $A,B$ two tight submodules of
$M$. Consider the following conditions:

(i) $A\cap B$ is tight in $A$;

(ii) $A\cap B$ is tight in $B$;

(iii) $A$ is tight in $A+B$;

(iv) $B$ is tight in $A+B$;

(v) $A\cap B$ is tight in $M$;

(vi) $A\cap B$ is tight in $A+B$;

(vii) $A+B$ is tight in $M$.

Then conditions (i)-(v) are equivalent, (vi) follows from each of them, and
(vii) implies each of them.

###### Proof.

(i) $\Leftrightarrow$ (iv) as well as (ii) $\Leftrightarrow$ (iii) follows
from Noether’s isomorphism theorem.

(i) $\Leftrightarrow$ (v). The third non-zero module in the exact sequence

$0\to A/(A\cap B)\to M/(A\cap B)\to M/A\to 0$

has p.d.$\leq 1.$ We deduce, by the well-known Kaplansky’s lemma on p.d.’s in
short exact sequences (see e.g. [15, Lemma 2.4, p. 202]) that the first two
non-zero modules in the last exact sequence are simultaneously of p.d. $\leq
1$.

(vii) $\Rightarrow$ (ii). From the exact sequence

$0\to(A+B)/A\to M/A\to M/(A+B)\to 0$

where $M/A$ and $M/(A+B)$ have p.d.$\leq 1$ we argue, by the same Kaplansky’s
lemma that p.d.$(A+B)/A=1$. Hence p.d.$B/(A\cap B)=1$, and $A\cap B$ is tight
in $B$.

(ii) $\Rightarrow$ (v). By hypothesis $A\cap B$ is tight in $B$, and $B$ is
tight in $M$. Therefore, by Lemma 2.1(a), $A\cap B$ is tight in $M$. Similar
argument yields (i) $\Rightarrow$ (v).

(vii) $\Rightarrow$ (iii). Because of Lemma 2.1(b), $A$ and $A+B$ tight in $M$
implies $A$ is tight in $A+B$.

(i)+(ii) $\Rightarrow$ (vi). If $A/(A\cap B)$ and $(A+B)/A\cong B/(A\cap B)$
have p.d.$\leq 1$, then the middle term in the exact sequence

$0\to A/(A\cap B)\to(A+B)/(A\cap B)\to(A+B)/A\to 0$

is likewise of p.d.$\leq 1$. ∎

We are indebted to L. Salce for furnishing us with an example showing that the
sum of two tight submodules of a module of p.d.1 need not be tight even if
their intersection is tight (the converse is ruled out by the implication
(vii) $\Rightarrow$ (v) in Lemma 2.2). Since the proof requires several
results from the theory of valuation domains that are not needed in this
paper, we skip the details.

The relation between the module and its tight submodules is a fundamental
issue. The following simple fact might provide useful information.

######  Lemma 2.3.

Let $M$ be a $V$-module of p.d.$\leq 1$. A tight submodule $C$ of $M$
satisfies $\mathop{\rm gen}C\leq\mathop{\rm gen}M.$

###### Proof.

If $\mathop{\rm gen}M=n$ is an integer, then by Warfield [21] $M$ is the
direct sum of $n$ cyclic submodules, so its Goldie-dimension is $n$ (the
Goldie-dimension — to be abbreviated as Gd — of a module $M$ is the supremum
of the cardinalities of the sets of non-zero summands in direct sums
$\oplus_{i\in I}M_{i}$ contained in $M$). A submodule $C$ cannot have larger
Goldie-dimension, thus $\mathop{\rm gen}C\leq n$.

$\begin{CD}&&&&0&&0\\\ &&&&@V{}V{}V@ VVV\\\ &&&&H@=H\\\ &&&&@V{}V{{}}V@
VV{}V\\\ 0@>{}>{}>C@>{}>{}>N@>{}>{}>F@>{}>{}>0\\\
&&||&&@V{}V{}V@V{}V{\psi}V\\\ 0@>{}>{}>C@>{}>{}>M@>{{\phi}}>{}>M/C@>{}>{}>0\\\
&&&&@V{}V{}V@ VVV\\\ &&&&0&&0\end{CD}$

If $\mathop{\rm gen}M={\kappa}$ is an infinite cardinal, then consider a free
resolution $0\to H\to F\to M/C\to 0$ where $\mathop{\rm gen}F={\kappa}$. The
pullback $N$ of ${\phi}:M\to M/C$ and $\psi:F\to M/C$ fits in the vertical
exact sequence $0\to H\to N\to M\to 0$. (See the commutative diagram with
exact rows and columns.) Obviously, the free submodule $H$ of the free $F$
satisfies $\mathop{\rm gen}H\leq\mathop{\rm gen}F={\kappa}$, whence
$\mathop{\rm gen}N\leq\mathop{\rm gen}H+\mathop{\rm gen}M={\kappa}$ and
$\mathop{\rm gen}N\geq\mathop{\rm gen}M={\kappa}$ follow. Thus $\mathop{\rm
gen}N={\kappa}$, and as $N\cong F\oplus C$, the inequality $\mathop{\rm
gen}C\leq\mathop{\rm gen}N={\kappa}$ becomes evident. ∎

Next we verify a result on the tightness of modules in a chain. Keep in mind
that being tight is not an inductive property. $($Continuity in the next lemma
means that $M_{\rho}=\cup_{{\sigma}<{\rho}}M_{\sigma}$ whenever ${\rho}$ is a
limit ordinal.$)$

######  Lemma 2.4.

Suppose

$None$ $0=M_{0}<M_{1}<\dots<M_{\sigma}<\dots<M_{\tau}=M$

is a continuous well-ordered ascending chain of submodules of the module $M$
with union $M$ such that the modules $M_{\sigma}\ ({\sigma}<{\tau})$ have
p.d.$1$. If each module is tight in its immediate successor, then each module
is tight in all of the successor modules of the chain, and also in $M$.
Furthermore, p.d.$M=1$.

###### Proof.

Apply Auslander’s familiar criterion on the p.d. of the union of a chain (see
e.g. [15, Chap. VI, Lemma 2.6]) to the subchain starting at $M_{\sigma}$, to
argue that p.d.$M_{{\sigma}+{\rho}+1}/M_{{\sigma}+{\rho}}\leq 1$ for all
ordinals ${\rho}\geq 0$ implies that $M_{\sigma}$ is tight in every successor
in the chain. That p.d.$M=1$ follows from the same lemma. ∎

Let us agree that when we say “object”, we will mean an object of the subclass
${\mathcal{C}}_{V}^{*}$ of ${\mathcal{C}}_{V}$ (remember: all objects have
p.d.$\leq 1$, and all subobjects are tight!). Manifestly,
${\mathcal{C}}_{V}^{*}$ is closed under direct summands and direct sums; the
projections onto direct summands and the embeddings of direct summands are
morphisms in ${\mathcal{C}}_{V}^{*}$. In the opposite direction, we observe
that neither the sum nor the intersection of two subobjects is necessarily a
subobject, and the union of an ascending chain of objects need not be an
object. The class ${\mathcal{C}}_{V}^{*}$ is not additive in general; it is of
course if $V$ is a discrete valuation domain (DVD).

##  3\. Fundamental Objects

Because of the restrictive composition rule of morphisms in our class, we have
to learn from scratch if and how some of the familiar basic concepts have to
be modified to fit in. One should not be surprised if in some cases hard-to-
believe situations have to be accepted.

Cyclically presented objects. The simplest objects in our class
${\mathcal{C}}_{V}^{*}$ are the cyclically presented modules: $Vr/Vs$ ($r,s\in
V$) where $Vs\leq Vr$ are principal ideals. All cyclic submodules of $Vr/Vs$
are subobjects. Later on we will see that these are the only subobjects of
$Vr/Vs\cong V(rs^{-1})\leq V$. Accordingly, only the principal ideals are
subobjects in object $V$.

Multiplications by ring elements induce endomorphisms of $Vr/Vs$ whose kernels
and images are subobjects. Morphisms between cyclically presented modules are
the same in ${\mathcal{C}}_{V}^{*}$ as in ${\mathcal{C}}_{V}$, therefore the
$V$\- as well as the ${\mathcal{C}}_{V}^{*}$-endomorphism ring of a cyclically
presented object is local.

Finitely generated objects. The finitely generated objects behave very nicely
in ${\mathcal{C}}_{V}^{*}$. It is an elementary result that a finitely
generated module over a valuation domain is of p.d.$\leq 1$ if and only if it
is finitely presented (see [14, Proposition 4.1, p. 83]), and then it is the
direct sum of a finite number of cyclically presented modules (see Warfield
[21], or [15, p.159]). Thus in our class ${\mathcal{C}}_{V}^{*}$, the finitely
generated objects are the finitely presented $V$-modules. Note that the
annihilators of elements in such a module are principal ideals (the ideal 0 is
included), thus subobjects.

######  Example 3.1.

Let $\\{x,y,z,w\\}$ be a generating set of the module $M$ with defining
relations $rx=0,\ sy=0,\ ax+by+cz=0,ex+dw=0$ where $r,s,a,b,c,d,e$ are non-
units in $V^{\times}$. The structure of $M$ depends to a great extent on the
divisibility relations between these ring elements. Assume e.g. the following
proper divisibility relations: $c\mid d\mid b\mid e\mid a\mid s\mid r$. We can
choose a different generating set: $\\{x,y,z^{\prime},w^{\prime}\\}$ with
$z^{\prime}=z+ac^{-1}x+bc^{-1}y,\ w^{\prime}=w+ed^{-1}x$; then the new
relations will be $rx=0,\ sy=0,\ cz^{\prime}=0,\ dw^{\prime}=0$. This shows
that the finitely presented module $M$ is the direct sum of four cyclically
presented submodules generated by $x,y,z^{\prime},w^{\prime}$ with
annihilators $Vr,Vs,Vc,Vd$, respectively. ∎

If $B$ is a finitely generated submodule of a finitely presented module $A$,
then $A/B$ is finitely presented, so it has p.d.$\leq 1$. Thus $B$ is a
subobject in $A$. If $C$ is tight in a finitely presented module $A$, then
p.d.$A/C\leq 1$ implies that $A/C$ is finitely presented. Hence $C$ is
finitely generated. Therefore, the subobjects in a finitely generated object
are precisely the finitely generated submodules. An immediate consequence of
this fact is the following corollary. It is telling us that the subclass of
finitely presented objects behaves the same manner in ${\mathcal{C}}_{V}^{*}$
as in ${\mathcal{C}}_{V}$.

######  Corollary 3.2.

1) Let ${\alpha}:A\to B$ and ${\beta}:B\to C$ be morphisms between finitely
presented objects in ${\mathcal{C}}_{V}^{*}$. Then the composite map
${\beta}{\alpha}$ is also a morphism in ${\mathcal{C}}_{V}^{*}$.

2) The endomorphisms of a finitely presented object in ${\mathcal{C}}_{V}^{*}$
form a ring.

###### Proof.

It suffices to show that the intersection of two finitely generated subobjects
is also finitely generated (the sum of two finitely generated submodules is
evidently again finitely generated). The implication (vii)$\Rightarrow$(vi) in
Lemma 2.2 ensures that such an intersection is tight also in the sum of the
objects. Tight in finitely generated is finitely generated. ∎

Important observations on finitely generated submodules in objects are
recorded in the following proposition.

######  Proposition 3.3.

(i) Finitely generated submodules in any object are $($finitely presented$)$
subobjects. In a finitely generated object they are the only subobjects.

(ii) The finitely presented subobjects of an object $M$ in the class
${\mathcal{C}}_{V}^{*}$ form a sublattice in the lattice of submodules of $M$.

(iii) The sum and intersection of a finitely presented subobject with any
subobject are also subobjects.

###### Proof.

(i) This is an immediate consequence of [15, Lemma 6.4, p. 217], which ensures
that finitely generated submodules of modules of p.d.$\leq 1$ are tight. The
second part of claim (i) was already stated above.

(ii) This follows from Corollary 3.2.

(iii) Let $A$ be a finitely presented and $B$ an arbitrary subobject of $M$.
The module $(A+B)/B$ is a finitely generated submodule of $M/B$, therefore, it
is tight in $M/B$. Hence $A+B$ is tight in $M$. The rest follows from Lemma
2.2. ∎

Uniserial objects. Most important objects are the uniserial modules (also
called serial modules in the literature): these are defined as modules whose
submodules form a chain with respect to inclusion. The obvious examples in
${\mathcal{C}}_{V}$ are the so-called standard uniserial modules: the field of
quotients, $Q$, as well as its submodules and submodules of their epic images,
i.e. modules of the form $J/I$ where $0\leq I<J\leq Q$ are submodules. By
Osofsky (see [18] or also [15, Chap. VI, Sect. 3]), the p.d. of a submodule
$J$ of $Q$ over a valuation domain is an integer $n\geq 0$ if and only if gen
$J={\aleph}_{n-1}$ (${\aleph}_{-1}$ means “finite”). Hence uniserial objects
in ${\mathcal{C}}_{V}^{*}$ are at most countably generated. However, — as
noted before — a countably generated ideal $J$ is not a subobject of $V$ in
${\mathcal{C}}_{V}^{*}$; indeed, as p.d.$V=0$ and p.d.$J=1$ imply p.d.$V/J=2$.
A torsion uniserial ought to have p.d.$\leq 1$ to belong to
${\mathcal{C}}_{V}^{*}$, therefore only those standard uniserials $J/I$ are
objects for which $J$ is at most countably generated and $I$ is cyclic, since
p.d.$J/I\leq 1$ implies that $J/I$ is coherent; see e.g. [14, Chap. IV,
Theorem 4.3]. Thus all the annihilator ideals of elements in a uniserial
object are principal ideals. Hence it follows that the only proper subobjects
of uniserial objects are cyclically presented. The non-standard uniserials
(that play an important role in the theory of valuation domains, cf. [2] or
[15]) will now be ignored, since their p.d. always exceeds 1.

A convenient way to deal with uniserial objects is to view them in the form
$J$ (torsion-free case) or $J/V$ (torsion case), where $J$ is either $Q$
(provided it is countably generated) or an at most countably generated proper
submodule of $Q$ containing $V$. Then it is trivial to answer the question of
isomorphism of uniserial modules: $J\cong J^{\prime}$ if and only if
$J=rJ^{\prime}$ or $J^{\prime}=rJ$ for some $r\in V^{\times}$ (i.e.
$J^{\prime}=qJ$ for some $0\neq q\in Q$), while $J/V\cong J^{\prime}/V$ if and
only if $J=J^{\prime}$. Also, it makes sense to talk about total order even in
the set of torsion uniserial objects in ${\mathcal{C}}_{V}^{*}$: the order
relation being induced by the natural inclusion relation of the numerators
$J$.

Let $U$ be a uniserial object and $r\in V^{\times}$. It is obvious what is
meant by $rU$. But it also makes sense to write $r^{-1}U.$ Indeed, this
denotes the uniserial object $U^{\prime}$ that satisfies $rU^{\prime}=U$; it
is unique up to isomorphism.

From the description of the proper subobjects of a countably generated
uniserial object $U\in{\mathcal{C}}_{V}^{*}$ it follows that their only
endomorphisms are the automorphisms and the map to the zero submodule, that
is, $\mathop{\rm End}\nolimits_{{\mathcal{C}}_{V}^{*}}(U)=\mathop{\rm
Aut}\nolimits_{V}(U)\cup\\{0\\}$. It is well known that the endomorphism ring
of a uniserial module in ${\mathcal{C}}_{V}$ is a local ring (see Shores–Lewis
[19]), and modules with local endomorphism rings enjoy the Exchange Property.

The Exchange Property is one of the properties most frequently investigated
about the behavior of summands. Recall that a module $A$ has the (finite)
Exchange Property if direct decompositions $M=A\oplus B=C\oplus D$ of any
module $M$ imply that there is another decomposition of the form $M=A\oplus
C_{1}\oplus D_{1}$ such that $C_{1}\leq C,D_{1}\leq D$. Furthermore, $A$ has
the Cancellation Property if $A\oplus B\cong A\oplus C$ for arbitrary modules
$B,C$ implies $B\cong C$. Finally, $A$ enjoys the Substitution Property if
$M=A_{1}\oplus B=A_{2}\oplus C$ with $A_{1}\cong A_{2}\cong A$ implies the
existence of a submodule $A^{\prime}\leq M$ such that $A^{\prime}\cong A$ and
$M=A^{\prime}\oplus B=A^{\prime}\oplus C$. (See [6].) We note that, if an
object $A\in{\mathcal{C}}_{V}^{*}$ has either the Exchange, or the
Cancellation, or the Substitution Property as a module in ${\mathcal{C}}_{V}$,
then it also displays the same property in the class ${\mathcal{C}}_{V}^{*}$,
since this class is closed under direct summands and direct sums.

In view of the preceding remarks, from [15, Corollary 2.3, p. 342] and [15,
p.181] we derive the following theorem.

######  Theorem 3.4.

Finite direct sums of uniserial objects in the class ${\mathcal{C}}_{V}^{*}$
enjoy all of the cancellation, exchange and substitution properties. ∎

Maximal uniserial submodules. By a maximal uniserial submodule in $M$ we mean
a uniserial submodule that is not properly contained in another uniserial
submodule of $M$.

######  Theorem 3.5.

Suppose $M$ is a $V$-module of p.d.$\leq 1$, and $U$ is a uniserial submodule
in $M$.

(i) If $U$ is maximal uniserial in $M$, then it is at most countably generated
and has p.d.$\leq 1$.

(ii) If $U$ is countably generated and tight in $M$, then it is a maximal
uniserial submodule in $M$.

(iii) If $U$ is a countably generated maximal uniserial submodule in a tight
submodule of $M$, then it is also maximal in $M$.

###### Proof.

(i) See [15, Chap. VI, Lemma 6.7].

(ii) By way of contradiction assume $U$ is not maximal in $M$, i.e. there
exists a uniserial $U^{\prime}\leq M$ that contains $U$ properly. We may
assume that $U^{\prime}$ is cyclic, say, $U^{\prime}=Va$ for $a\in M$. Then
$Va/U$ is a non-zero cyclic submodule in the module $M/U$ which has p.d.1 by
hypothesis. Hence it follows that $Va/U$ is tight in $M/U$, so $U$ must be
cyclic. This contradiction completes the proof of (ii).

(iii) The proof is similar to that of (ii). If $U$ is a maximal uniserial in a
tight submodule $N$ of $M$ and contained in a larger uniserial $U^{\prime}\leq
M$ that is cyclic, then $U^{\prime}/U\cong(U^{\prime}+N)/N$ is a cyclic
submodule in $M/N$, so cyclically presented. Again we can conclude that $U$
must be cyclic. ∎

An immediate consequence of this theorem is that in a direct sum of cyclically
presented objects all uniserial subobjects are also cyclically presented.

Mixed modules as objects. An object in ${\mathcal{C}}_{V}^{*}$ that is mixed
in the usual sense (i.e. neither torsion nor torsion-free) need not have a
1-dimensional torsion part. Actually, the torsion submodule of a mixed module
of p.d.1 can have any p.d. not exceeding the maximal p.d. of torsion-free
$V$-modules minus 1 whenever this number is $\geq 1$. Indeed, select a
torsion-free $V$-module $N$ of p.d. $n\geq 2$ and set $N=F/G$ with a free
$V$-module $F$. Let $H$ be an essential free submodule of $G$, and define
$M=F/H$. Then $M$ is an object with torsion submodule $G/H$ of p.d. $n-1$.

Even if the torsion submodule of a mixed object is an object, it need not be a
subobject. Therefore, it seems reasonable to consider an object mixed in
${\mathcal{C}}_{V}^{*}$ if its torsion submodule is a non-zero proper
subobject. For an injective object in ${\mathcal{C}}_{V}^{*}$ that is a mixed
module in ${\mathcal{C}}_{V}$, but not in ${\mathcal{C}}_{V}^{*}$, see Example
6.9 infra.

Countably generated objects. Suppose $M$ is the union of a countable ascending
chain $M_{n}\ (n<{\omega})$ of finitely presented modules. Then each $M_{n}$
is tight in its immediate successor and hence also in $M$ which will have
p.d.$1$ (cf. Lemma 2.4). Moreover, since every finitely generated submodule of
$M$ is contained in some $M_{n}$, all finitely generated submodules of $M$ are
tight in $M$ and finitely presented, so subobjects. Cyclic subobjects are
cyclically presented, thus the annihilators of elements in $M$ are principal
ideals (i.e. also objects). Submodules that are countably generated may or may
not be subobjects in countably generated objects. E.g. the countably generated
ideals of $V$ are objects, but not subobjects of $V$.

Claim (i) in our next theorem shows that the module $M$ in the preceding
paragraph is a typical countably generated object.

######  Theorem 3.6.

(i) A countably generated $V$-module is of p.d.$\leq 1$ if and only if it is
the union of a countable ascending chain of finitely presented $($tight$)$
submodules.

(ii) Let $A,B$ be tight submodules in a $V$-module $M$ of p.d.$\leq 1$ such
that $B$ is at most countably generated. Then

(a) $C=A\cap B$ is also at most countably generated, and tight in $B$ and also
in $A$ and in $M$;

(b) $A+B$ is of p.d.$\leq 1$, and $A$ is tight in it.

###### Proof.

(i) One way the claim follows from Proposition 3.3, while the converse is
taken care of by the last but one paragraph before this theorem.

(ii) (a) $B$ is the union of a countable chain $\\{B_{n}\ |\ n<{\omega}\\}$ of
finitely presented submodules. Hence $B/C=B/(A\cap
B)\cong(A+B)/A=\cup_{n}(A+B_{n})/A$, where the last quotient modules are
finitely generated (epic images of the $B_{n}$ in $M/A$), and therefore tight
in $M/A$. The p.d. of their union $(A+B)/A$ equals 1, which means p.d.$B/C=1$.
Thus $C$ is tight in $B$, and hence in $M$, and then also in $A$.

(b) The sum $A+B$ is the union of the chain of the extensions $A+B_{n}$ of $A$
by finitely presented modules, so its p.d. is $\leq 1$. ∎

Next we give more examples of countably generated objects. By choosing a
suitable value group, it is easy to construct valuation domains that have the
required properties.

######  Example 3.7.

Consider a uniserial object $U$ generated by $\\{u_{i}\ (i<{\omega})\\}$.
Attach to each $u_{i}\in U$ a cyclically presented module, say, generated by
$b_{i}$ such that $r_{i}b_{i}=u_{i}$ where the non-units $r_{i}\in V^{\times}$
are required to satisfy the condition that $V/Vr_{i}$ is properly embeddable
in $U/Vu_{i}$. Then $U$ is a pure subobject of $B=\langle U,b_{i}\
(i<{\omega})\rangle$ such that $B/U$ is a direct sum of cyclically presented
modules $\cong Vb_{i}/Vu_{i}$. Since pure extensions by a cyclically presented
module are obviously splitting, we obtain: $B\cong
U\oplus(\oplus_{i<{\omega}}Vb_{i}/Vu_{i})$. ∎

######  Example 3.8.

Let $J,L$ denote submodules of $Q$ containing $V$ that are at most countably
generated, and let $r\in V^{\times}$ be a non-unit. Then one of $J/Vr$ and
$L/Vr$, say, the former, admits an isomorphic embedding in the latter,
${\phi}:J/Vr\to L/Vr$, that is the identity on $V/Vr$. Define $N$ as the push-
out of the embeddings ${\phi}:V/Vr\to J/Vr$ and $\psi:V/Vr\to L/Vr$. Then $N$
is an extension of its pure submodule $L/Vr$ by $J/V$. A fast calculation
shows that $N=L/Vr\oplus J^{\prime}/V$ where $J^{\prime}/V=\\{(x,{\phi}(x))\
|\ x\in J/V\\}\leq J/V\oplus L/V$. ∎

The abundance of subobjects. It is a remarkable fact that even if $V$ is not a
discrete rank one valuation domain, objects in ${\mathcal{C}}_{V}^{*}$ contain
a large number of tight submodules of all possible sizes. From our discussion
above this is clear for finitely and countably generated objects, while for
uncountably generated objects it is a consequence of the existence of tight
systems. Every module $M$ of p.d.$\leq 1$ admits a tight system
${\mathcal{T}}$ (over any integral domain). This is defined as a
$G({\aleph}_{0})$-family of tight submodules; see [15, Chap. VI, Sect. 5].
Recall that, for an infinite cardinal ${\kappa}$, by a $G({\kappa})$-family
${\mathcal{G}}$ of submodules of a module $M$ is meant a set of submodules
such that the following conditions are satisfied: 1) $0,M\in{\mathcal{G}}$; 2)
${\mathcal{G}}$ is closed under unions; 3) if $X$ is a subset of $M$ of
cardinality $\leq{\kappa}$ and $A\in{\mathcal{G}}$, then there exists a
$B\in{\mathcal{G}}$ such that $X\cup A\subseteq B$ and gen$(B/A)\leq{\kappa}$.
It follows that in case ${\mathcal{T}}$ is a tight system of $M$, then $A<C\
(A,C\in{\mathcal{T}})$ implies $A$ is tight in $C$. Moreover, under the
canonical map the complete preimages of the tight submodules in $C/A$ are
subobjects of $C$; they are also subobjects in $M$.

An immediate corollary of the existence of tight systems is the next result.

######  Corollary 3.9.

Let $M$ be an object, and $N$ a submodule of $M$. Then $N$ is contained in a
tight submodule $N^{*}$ of $M$ such that $\mathop{\rm
gen}N^{*}\leq\max\\{\mathop{\rm gen}N,{\aleph}_{0}\\}$.

###### Proof.

Every generator of $N$ is contained in some countably generated tight
submodule that belongs to a fixed tight system ${\mathcal{T}}$ of $M$. The
union of all these members of ${\mathcal{T}}$ is a member $N^{*}$ of
${\mathcal{T}}$ containing $N$. By construction, $N^{*}$ can be generated by
$\mathop{\rm gen}N\cdot{\aleph}_{0}$ elements. ∎

A tight system ${\mathcal{T}}$ of object $M$ allows us to build a continuous
well-ordered ascending chain (1) of tight submodules
$M_{\sigma}\in{\mathcal{T}}$ for some ordinal ${\tau}$, such that
gen($M_{{\sigma}+1}/M_{\sigma})\leq{\aleph}_{0}$ for all ${\sigma}<{\tau}$.
Moreover, since a countably generated object is the union of a chain of
finitely presented subobjects (Theorem 3.6), the chain (1) can be refined so
as to have all of the quotients $M_{{\sigma}+1}/M_{\sigma}$ finitely, or even
cyclically presented.

Another important consequence of the existence of tight systems is that we
have already formulated in Proposition 3.3: every finitely generated submodule
of a module $M$ (of any size) of p.d.$1$ is finitely presented and tight in
$M$.

Projective objects. A finitely generated torsion-free module over a valuation
domain is free. A useful fact: a finite rank pure submodule in a free
$V$-module is a summand; see e.g. [14, Chap. XIV, Theorem 6.1]. By Kaplansky
[16], projective $V$-modules are free. Evidently, they are projective objects
in our class ${\mathcal{C}}_{V}^{*}$ as well. Dimension calculation shows that
a tight submodule of a free module must have zero p.d. Therefore, we can
state:

######  Theorem 3.10.

The subobjects of free $V$-modules are the free submodules. ∎

Hence we conclude that every object $M$ in ${\mathcal{C}}_{V}^{*}$ admits a
free resolution in the form of a short exact sequence: $0\to H\to F\to M\to 0$
where $H,F$ are free $V$-modules, i.e. free objects.

##  4\. More Fundamental Concepts

Continuing the review of the basics, we would like to establish more results
concerning the objects in the class ${\mathcal{C}}_{V}^{*}$, but in order to
deal with the objects more efficiently, we need several tools available in
[14] and in [15]. In this section we review some concepts and facts we shall
need.

Annihilators of elements. The study of objects in ${\mathcal{C}}_{V}^{*}$ is
greatly simplified by the fact that the annihilator ideals of elements in
objects are not just objects, but they are even principal ideals. This has
been pointed out before, but let us give a formal proof of this property.

######  Lemma 4.1.

Let $M$ be an object in ${\mathcal{C}}_{V}^{*}$. Then for any element $a\in
M$, the annihilator $\mathop{\rm ann}\nolimits_{M}(a)=\\{r\in V\ |\ ra=0\\}$
is a principal ideal of $V$.

###### Proof.

$M$ has a tight system, so $a\in M$ is included in a countably generated tight
submodule $N$ of $M$, and hence also in a finitely presented submodule (see
Theorem 3.6 above). For finitely presented objects the claim has been
established before. ∎

Heights of elements. The principal information in describing the way an
element is located in the module is stored in its height. Heights of elements
are defined by using uniserial modules, see [14]. The uniserials that occur as
possible heights for valuation domains have been studied in [1] and [2].
Fortunately, in modules of p.d.$\leq 1$ only most tractable heights can occur.

Suppose $M$ is an object and $0\neq a\in M$. Consider maps ${\phi}_{J}:J\to M$
of the submodules $J$ of $Q$ containing $V$ such that ${\phi}_{J}(1)=a$. For a
fixed $a$, the union in $Q$ of those $J$’s for which such a ${\phi}_{J}$
exists is a submodule $H_{M}(a)$ of $Q$, called the height-ideal of $a\in M$.
The module

$h_{M}(a)=H_{M}(a)/V$

is defined as the height of $a\in M$. We call $h_{M}(a)$ non-limit height or
limit-height according as $H_{M}(a)$ is one of the $J$’s or is not. In the
limit case we write $h_{M}(a)=U^{-}$. Note that $h_{M}(a)$ is always a
uniserial torsion module; it is of the form $U=J/V$ with $J\subseteq Q$
(equality only in case $Q\in{\mathcal{C}}_{V}^{*}$). In the non-limit case,
the element $a$ is contained in a uniserial module $W$ that is a maximal
uniserial in $M$ such that $h_{M}(a)=W/Va$. The heights of elements in a non-
standard uniserial are uncountable limit heights — these are out of question
in ${\mathcal{C}}_{V}^{*}$. The set of heights occurring in
${\mathcal{C}}_{V}^{*}$ is totally ordered in the obvious way once we declare
the non-limit height $J/V$ to be larger than the corresponding limit height
$(J/V)^{-}$. The minimum height is 0 (this is the height of the generator in a
cyclic module), and we set $h(0)=\infty$ as the maximum height.

######  Example 4.2.

To give an example of a limit height, consider a countably generated submodule
$J$ of $Q$ containing $V$, and choose a properly ascending chain of fractional
ideals $\\{Vt_{i}^{-1}\ |\ i<{\omega}\\}$ with union $J$ (where $t_{i}\in
V^{\times}$). Define a countably generated object $X$ as follows: the
generators are $x_{i}\ (i<{\omega})$ with the defining relations:

$None$ $rt_{0}x_{0}=0,\quad t_{0}x_{0}=t_{i}x_{i}\qquad(i<{\omega})$

where $r\in V^{\times}$ is arbitrary. The element $t_{0}x_{0}$ has limit
height, namely $(J/r^{-1}V)^{-}$. To get an idea of what kind of module $X$
is, observe that the cyclic submodules generated by the elements
$x_{i}-(t_{i}^{-1}t_{i+1})x_{i+1}$ are summands of $X$ for all $i<{\omega}$ (a
complement is the submodule generated by all the given generators with $x_{i}$
removed). Actually, these cyclic modules generate their direct sum
$X^{\prime}$ in $X$. This $X^{\prime}$ is tight and pure in $X$, and the
quotient $X/X^{\prime}$ is a countably generateduniserial module containing
the coset $x_{0}+X^{\prime}$. ∎

Next we prove the following:

######  Theorem 4.3.

A non-zero element in an object of the class ${\mathcal{C}}_{V}^{*}$ has one
of the following heights:

(i) cyclic height;

(ii) countably generated non-limit height;

(iii) arbitrary limit height of standard type.

Elements in a finitely generated module cannot have limit heights.

###### Proof.

To begin with, observe that for each of (i)-(iii) we already had examples
above, so it remains only to show that (i)-(iii) is a complete list. The only
other heights in ${\mathcal{C}}_{V}$ are uncountably generated non-limit
heights. Working toward contradiction, suppose that for some $a\in
M\in{\mathcal{C}}_{V}^{*}$, we have $h_{M}(a)=J/V$ with an uncountably
generated submodule $J$ of $Q$. There is a homomorphism ${\phi}_{J}:J\to M$
such that ${\phi}_{J}(1)=a$. The maximal property of $J$ as height implies
that ${\phi}_{J}(J)$ must be a maximal uniserial in $M$. Therefore, by Theorem
3.5 it is at most countably generated — a contradiction, completing the proof
of the first claim.

That (iii) cannot occur in a finitely generated module is an immediate
consequence of the simple fact that limit heights require infinite Goldie-
dimension, as is clear from Example 4.2. ∎

Height-gaps. Suppose $U$ is a uniserial object and $Vr\neq 0$ is the
annihilator of $a\in U$. Then $h_{U}(a)=rU$ and $h_{U}(sa)=s^{-1}rU$ provided
that $sa\neq 0$ for $s\in V$. If $U$ is contained in a $V$-module $M$, then
the heights of these elements may be larger in $M$. In general, in every
module $M$, for an element $a\in M$ and its multiple $ra\ (r\in V^{\times})$
the inequality

$h_{M}(a)\leq r^{-1}h_{M}(ra)$

holds. We say that $M$ has a height-gap at $0\neq a\in M$ if,
$h_{M}(a)>sh_{M}(x)$ holds whenever $sx=a$ for some $x\in M$ and for a non-
unit $s\in V$.

######  Example 4.4.

To illustrate height-gaps, let $U$ be a uniserial module, and
$x_{1},x_{2},x_{3}$ symbols. Suppose that the non-units $s_{i},t_{i}\in
V^{\times}$ satisfy the following proper divisibility relations: $s_{1}\mid
s_{2}\mid s_{3}$ and $t_{1}\mid t_{2}\mid t_{3}$. Pick some $u\in U$ such that
$s_{3}u\neq 0$, and define a module $N$ to be generated by $U$ and by the
given symbols subject to the relations $s_{i}u=s_{i}t_{i}x_{i}\ (i=1,2,3).$
The height-gaps in the submodule $U$ are at $s_{1}u,s_{2}u,s_{3}u$, and at
$0$. ∎

Purity. The main point about this widely used concept that we are emphasizing
repeatedly is that in valuation domains it is equivalent to the simpler
relative divisibility (see [20]). Thus a submodule $N$ is pure in a $V$-module
$M$ if and only if $rN=N\cap rM$ holds for every $r\in V^{\times}$.
Equivalently, for all $r\in V$, the map $V/Vr\otimes_{V}N\to V/Vr\otimes_{V}M$
induced by the inclusion $N\to M$ is monic. This is tantamount to the
injectivity of the map $\mathop{\rm Hom}\nolimits_{V}(V/Vr,M)\to\mathop{\rm
Hom}\nolimits_{V}(V/Vr,M/N)$ for all $r\in V^{\times}$ induced by the natural
homomorphism $M\to M/N$. A pure-exact sequence $0\to A\to B\to C\to 0$ is an
exact sequence in which the image of the map $A\to B$ is pure in $B$.

######  Lemma 4.5.

(i) Let $U$ be a uniserial submodule of an object $M$, and $a\in U$. If
$h_{M}(a)=U/Va$, then $U$ is a maximal uniserial in $M$, and there is no
height-gap in $U$ at $a$ and above.

(ii) If $U$ is a maximal uniserial in $M$ and is torsion with no height-gaps
other than the one at $0$, then $U$ is pure in $M$.

###### Proof.

(i) This is rather obvious.

(ii) $U$ is not pure in $M$ means that there is $u\in U$ such that
$h_{U}(u)<h_{M}(u).$ Hence there must be a height-gap in $U$ at $u$ or above,
because by maximality, some of the generators of $U$ have the same height in
$U$ as in $M$. ∎

We recall the definition of $\mathop{\rm Pext}\nolimits_{V}^{1}(X,M)$: it is a
sub-bifunctor of $\mathop{\rm Ext}\nolimits_{V}^{1}(X,M),$ consisting of those
non-equivalant extensions of $M$ by $X$ in which $M$ is a pure submodule (see
e.g. [15, p. 45]). In the commutative case, Pext is a $V$-module.

##  5\. Theorems on Torsion and Torsion-free Modules

In this section, we discuss briefly a few fundamental results on torsion and
torsion-free objects. An in-depth study that would require more research and
interesting applications is planned in the future.

We start with the following simple observation.

######  Theorem 5.1.

A pure and tight finitely generated submodule in an object is a direct
summand.

###### Proof.

Suppose a finitely generated module $N$ is pure and tight in a module $M$ of
p.d.$\leq 1$. By the tightness of $N$, $M/N$ has p.d.$\leq 1$, and by
Corollary 7.7 $N$ is pure-injective. All this combined implies that $N$ is a
summand of $M$. ∎

We continue with typical examples of direct sums of cyclically presented
modules: the pure-projective objects. These are defined as objects $P$ that
satisfy $\mathop{\rm Pext}\nolimits_{V}^{1}(P,M)=0$ for all objects
$M\in{\mathcal{C}}_{V}^{*}$.

######  Theorem 5.2.

A $V$-module is pure-projective if and only if it is a direct sum of
cyclically presented modules.

###### Proof.

This is a special case of a well-known theorem. E.g. it follows from [15,
Chap. VI, Theorem 12.2]. ∎

Concerning direct sums of uniserials, a most important result is the following
theorem (this is not related to tightness).

######  Theorem 5.3.

(i) The uniserial summands in a direct sum of uniserial modules are unique up
to isomorphism.

(ii) Summands of a direct sum of uniserial modules are themselves direct sums
of uniserials.

###### Proof.

These are well-known immediate consequences of the fact that the endomorphism
rings of uniserial modules are local (Theorem 3.4). ∎

Let $r\in V^{\times}$ be a non-unit. By a $V/Vr$-homogeneous module we mean a
$V$-module $H$ such that each element is contained in a submodule of $H$ that
is $\cong V/Vr$. Then $H$ satisfies $rH=0$, and any cyclic submodule of $H$
that is $\cong V/Vr$ must be pure in $H$. Moreover, by Theorem 5.1 it is then
a summand.

######  Proposition 5.4.

Suppose $M$ is a $V$-module of p.d.$1$.

(i) If $rM=0$, then a $V/Vr$-homogeneous tight submodule is a summand of $M$.

(ii) If $M$ is $V/Vr$-homogeneous, then it is the direct sum of cyclically
presented submodules, all isomorphic to $V/Vr$.

(iii) If $D$ is a divisible object, then for every $r\in V^{\times}$, $D[r]$
is $V/Vr$-homogeneous, so a direct sum of cyclically presented submodules
isomorphic to $V/Vr$.

###### Proof.

For (i)-(ii) we refer to [15, Chap. XII, Theorems 2.2 and 2.3], and for (iii)
to [15, Chap. XIV, Corollary 2.4]. ∎

We note that the number of summands $\cong V/Vr$ in (iii) is the same for
every $r$ provided that $D[r]\neq 0$: it is the Goldie-dimension of $D$.

An important theorem in abelian group theory, due to L. Ya. Kulikov, states
that a subgroup of a direct sum of cyclic groups is likewise a direct sum of
cyclic groups (see, e.g., [11]). An analogue in ${\mathcal{C}}_{V}^{*}$ would
state that a tight submodule of a direct sum of cyclically presented modules
is also such a direct sum. This is indeed true for torsion-free modules: a
tight submodule of a free module in ${\mathcal{C}}_{V}^{*}$ is again free. It
was conjectured that this holds in ${\mathcal{C}}_{V}^{*}$ also in the torsion
case. (For torsion abelian groups, see e.g. [11, Chap. 3, Theorem 5.7].)
However, we claim that the module $X$ in Example 4.2 refutes this conjecture.
In order to prove this, consider the module $Y$ in the following example.

######  Example 5.5.

The countably generated torsion object $Y$ is defined just as the module $X$
in Example 4.2: it is generated by the same set $\\{x_{i}\ |\ i<{\omega}\\}$
with the same defining relations, but there is a single modification: we
replace $t_{0}\in V^{\times}$ by $s\in V^{\times}$ that is picked such that
$J<Vs^{-1}$. In this case, $Vx_{0}$ is a pure and tight submodule in $Y$ (a
summand), and the elements $x_{i}-(t_{i}^{-1}s)x_{0}$ for all $i>0$ generate
cyclic direct summands of $Y$ such that $Y$ is the direct sum of $Vx_{0}$ and
these cyclic submodules. (Another, but less explicit argument to obtain the
structure of $Y$ is as follows. After observing that the cyclic submodule
$Vx_{0}$ is pure in $Y$, it only remains to point out that moreover, it is a
summand of $Y$, since $Y/Vx_{0}$ is pure-projective as the direct sum of
cyclically presented modules $\cong Vt_{i}\ (i>0)$.) ∎

To argue that the object $X$ of Example 4.2 cannot be a direct sum of
cyclically presented objects, appeal to Theorem 4.3. The element
$t_{0}x_{0}\in X$ is of countable limit height, and as such it cannot belong
to a direct sum of the stated kind: it would be contained already in a
finitely generated summand with the same limit height. However, this is
impossible as is demonstrated by the cited theorem. Thus the object $X$ that
is (isomorphic to) a tight submodule (observe that $Y/X\cong Vs/Vt_{0}$ is
cyclically presented) in a direct sum $Y$ of cyclically presented modules
fails to be a direct sum of such modules.

Hence it is obvious that this theorem of Kulikov cannot have the suspected
analogue in ${\mathcal{C}}_{V}^{*}$ without additional hypotheses. Looking for
simple conditions that would lead us to a Kulikov-type theorem for subobjects
in direct sums of cyclically presented objects, we selected (b), in addition
to the obvious (a), that seems natural to assume:

(a) The non-zero elements have cyclic heights.

(b) The uniserial submodules admit but a finite number of height-gaps.

Under the hypothesis of (a)-(b), we will prove a desired analogue for the
countably generated torsion objects (see Theorem 5.8 below). But first we deal
with preliminary lemmas.

We need a definition. Similarly to [13], we will call a $V$-module $M$
cyclically separable if every finite set of its elements can be embedded in a
finitely generated summand of $M$, i.e. in a summand that is the direct sum of
a finite number of cyclically presented modules. Observe that in order to
verify the cyclic separability of a torsion object, it suffices to check the
defining property only for one element subsets, as every finitely generated
object is a finite direct sum of cyclically presented objects. Hence it is
evident that summands of modules inherit cyclical separability. We now prove a
crucial lemma.

######  Lemma 5.6.

Let $M$ denote a torsion object in ${\mathcal{C}}_{V}^{*}$. If $M$ satisfies
conditions (a)-(b), then it is a cyclically separable $V$-module.

###### Proof.

Assume $M$ has properties (a)-(b), and let $0\neq a\in M$. By (a), $a$ is
contained in a cyclically presented submodule $C=Vc$ that is maximal uniserial
in $M$. If $C$ contains no height-gap strictly between $a$ and $0$, then $C$
is pure in $M$, and hence a summand of $M$ (Theorem 5.1). Thus in this case
$a$ embeds in a cyclically presented summand of $M$, and we are done. If there
are height-gaps in $C$ between $a$ and $0$, then by (b) there is one, say at
$rc\ (r\in V)$, such that no height-gap exists strictly between $rc$ and $0$.
Then by the previous argument there is a cyclically presented summand $B=Vb$
of $M$ that contains $rc$, $M=Vb\oplus M^{\prime}$. If $b^{\prime}\in Vb$ is
such that $Vrb^{\prime}/Vc\cong Vr$, then $V(c)+V(b)=V(c-b^{\prime})\oplus
V(b)$. In this case, the projection of $V(c-b^{\prime})$ in $M^{\prime}$
contains the coordinate of $a$ with a smaller number of height-gaps below it.
Repeating this process for the coordinates of $a$ a finite number of times, we
get a finitely generated summand of $M$ that contains the selected element
$a$. ∎

######  Lemma 5.7.

Let $M$ be a torsion object satisfying conditions (a)-(b). If $M$ is countably
generated, then it is a direct sum of cyclically presented objects.

###### Proof.

By Lemma 5.6, $M$ is cyclically separable. It is a simple exercise to prove
that a countably generated cyclically separable module is a direct sum of
cyclics. ∎

The following analogue of Kulikov’s theorem is now easily established.

######  Theorem 5.8.

Assume $M$ is a direct sum of cyclically presented torsion objects, and $N$ is
a countably generated subobject satisfying condition (b) above. Then $N$ is
likewise a direct sum of cyclically presented subobjects.

###### Proof.

Owing to Theorem 2.3 and Lemma 5.6 it suffices to show that $N$ satisfies
condition (a). But this is immediate by virtue of Theorem 3.5 (iii). ∎

We are asking the obvious question: do the preceding lemma and theorem hold
for larger cardinalities? The answer is: for Lemma 5.7 counterexamples are
torsion-complete abelian $p$-groups with countable unbounded basic subgroups;
cf. [11, Chap. 10, sect. 3]. For Theorem 5.8 we do not know the answer.

We record the following two parallel questions.

######  Problem 5.9.

Are pure subobjects in direct sums of torsion uniserial $($resp. countably
generated$)$ objects also direct sums of the same kind? ∎

Next we want to get an idea of the torsion-free modules in
${\mathcal{C}}_{V}^{*}$. It is a pleasant surprise that all countably
generated torsion-free modules in ${\mathcal{C}}_{V}$ are objects in
${\mathcal{C}}_{V}^{*}$. This is evident from the following theorem.

######  Theorem 5.10.

(i) A torsion-free $V$-module $A$ has p.d.$\leq 1$ if and only if every rank
one pure submodule is at most countably generated.

(ii) A torsion-free $V$-module $A$ is of p.d.$\leq 1$ if and only if it admits
a well-ordered ascending chain of tight pure submodules $A_{\alpha}$ such that
for each ${\alpha}$, $A_{{\alpha}+1}/A_{\alpha}$ is of rank one and of p.d.$1\
($thus cyclic or countably generated torsion-free$)$.

###### Proof.

It suffices to refer to [7, Corollary 4.5] and to [15, Chap. VI, Lemma 6.6],
respectively. ∎

We continue with a theorem that resembles Pontryagin’s theorem on countable
free abelian groups. A similar result for the projective dimension one case is
also included.

######  Theorem 5.11.

A torsion-free module of countable rank in ${\mathcal{C}}_{V}^{*}$ is free
$($is an object in ${\mathcal{C}}_{V}^{*})$ if and only if its finite rank
pure submodules are free $($have p.d.$\leq 1)$.

###### Proof.

See [15, Chap. VI, Corollary 3.12]. ∎

##  6\. Divisible and Injective Objects

The theory of divisibility and injectivity clearly illustrates a fundamental
difference between the classes ${\mathcal{C}}_{V}^{*}$ and
${\mathcal{C}}_{V}$.

Divisible objects. Divisibility of modules is defined as usual:
$D\in{\mathcal{C}}_{V}^{*}$ is divisible if $rD=D$ for all $r\in V^{\times}$.
Equivalently, the equality $\mathop{\rm Ext}\nolimits^{1}_{V}(V/Vr,D)=0$ holds
for all $r\in V^{\times}$. The prototype of divisible modules, the quotient
field $Q$ of $V$ as a $V$-module, is in general not an object. It is exactly
when $Q$ is a countably generated $V$-module (then p.d.$Q=1$, i.e. $V$ is a
Matlis domain). But the module $\partial_{V}$ (see [8]), the generator of the
subcategory of the divisible modules in $V$-Mod has p.d.1, so it is an object
in ${\mathcal{C}}_{V}^{*}$. Recall that $\partial_{V}$ is generated by the
$k$-tuples $(r_{1},\dots,r_{k})$ for all $k\geq 0$ of non-unit elements
$r_{i}\in V^{\times}$, subject to the defining relations

$None$ $r_{k}(r_{1},\dots,r_{k-1},r_{k})=(r_{1},\dots,r_{k-1})\qquad(k>0)$

for all choices of the $r_{i}$. The generator $w=(\emptyset)$ generates a
submodule of $\partial_{V}$ isomorphic to $V$ such that
$\partial_{V_{0}}=\partial_{V}/Vw$ is a divisible torsion module of p.d.1
(which is a generator of the subcategory of divisible torsion modules in
${\mathcal{C}}_{V}$). See [8] or [15] for more details.

As far as the structure of divisible objects is concerned, the following
information is crucial. (Pay attention to the enormous simplification over
Matlis domains.)

######  Theorem 6.1.

(i) An object in ${\mathcal{C}}_{V}^{*}$ is divisible if and only if it is a
summand of a direct sum of copies of the module $\partial_{V}$.

(ii) If $V$ is a Matlis domain, then an object is divisible if and only if it
is the direct sum of copies of $Q$ and/or $K$.

###### Proof.

(i) In [8, Theorem 18] it is shown that over a Prüfer domain (and hence over a
valuation domain) a divisible module has p.d.1 if and only if it is a summand
of a direct sum of copies of $\partial$. (By the way, this holds for all
integral domains.)

(ii) See [15, Chap. VII, Theorem 3.5]. ∎

In order to obtain a full set of invariants for a divisible object $D$, we
introduce two cardinal invariants measuring the size of its torsion and
torsion-free parts. One is ${\kappa}=\mathop{\rm rk}D$, the torsion-free rank
of $D$, the number of generators of a maximal size free submodule contained in
$D$. The other invariant is ${\lambda}=\mathop{\rm gen}D[r]$ for any non-unit
$r\in V^{\times}$. Thus ${\lambda}$ is the cardinality of the set of summands
$\cong V/Vr$ in a direct decomposition of $D[r]$ into indecomposable summands.
These two cardinals form a complete set of invariants characterizing divisible
objects in ${\mathcal{C}}_{V}^{*}$. In fact,

######  Theorem 6.2.

Assume $D$ and $D^{\prime}$ are divisible objects in ${\mathcal{C}}_{V}^{*}$.
Then $D\cong D^{\prime}$ if and only if

(i) their ranks are equal: $\mathop{\rm rk}D=\mathop{\rm rk}D^{\prime}$; and

(iI) for some, and hence for each $r\in V^{\times}$, $\mathop{\rm
gen}D[r]=\mathop{\rm gen}D^{\prime}[r]$.

###### Proof.

See [10, Theorem C] or [15, Chap. VII, Theorem 3.4]. ∎

We also state the existence theorem accompanying this structure theorem.

######  Theorem 6.3.

Given the cardinals ${\kappa},{\lambda}$, there exists a divisible object $D$
in class ${\mathcal{C}}_{V}^{*}$ such that $\mathop{\rm rk}D={\kappa}$ and
$\mathop{\rm gen}D[r]={\lambda}$ if and only if

(i) in case p.d.$Q=1$: both ${\kappa}$ and ${\lambda}$ are arbitrary;

(ii) in case p.d.$Q>1$: ${\kappa}$ is arbitrary and
${\lambda}\geq\max\\{{\kappa},\mathop{\rm gen}Q\\}.$

###### Proof.

We refer to [10, Theorem 3] or to [15, Chap. VII, Theorem 3.8]. ∎

From the foregoing results we can draw the conclusion that in case p.d.$Q>1$
every divisible $D\neq 0$ in ${\mathcal{C}}_{V}^{*}$ satisfies
Gd($D)\geq\mathop{\rm gen}Q$. Furthermore, no indecomposable divisible object
exists in ${\mathcal{C}}_{V}^{*}$.

We also have the embedding result as expected:

######  Theorem 6.4.

Every object in ${\mathcal{C}}_{V}^{*}$ is a subobject of a divisible object.

###### Proof.

Write $M\in{\mathcal{C}}_{V}^{*}$ as $M=F/H$ with free $V$-modules $F,H$. If
$F=\oplus_{i\in I}Vx_{i}$ with $Vx_{i}\cong V$, then define $G=\oplus_{i\in
I}\partial_{i}$ with $\partial_{i}\cong\partial_{V}$, and embed $F$ in $G$ by
identifying the generator $x_{i}$ with the generator $w_{i}\in\partial_{i}$,
for all $i$. Then $F$ becomes a subobject of $G$, and $G/H$ will be a
divisible module of p.d.1 that contains a copy of $M$ as a subobject. ∎

$h$-divisibility of a $V$-module $H$ is defined by the extendibility of the
homomorphisms $V\to H$ to $Q\to H$ (see [17] or [15, p. 38]). With the
exception of the next proposition, this concept will not be discussed,
considering that $h$-divisible modules rarely exist in
${\mathcal{C}}_{V}^{*}$, even injective objects are not $h$-divisible whenever
p.d.$Q>1$.

######  Proposition 6.5.

(i) $h$-divisible objects exist in ${\mathcal{C}}_{V}^{*}$ if and only if $V$
is a Matlis domain, in which case all divisible modules are $h$-divisible.

(ii) If $V$ is a Matlis domain, then an $h$-divisible object of p.d.$1$ is the
direct sum of copies of $Q$ and $K$.

###### Proof.

(i) It is well known that over a domain, divisibility and $h$-divisibility are
equivalent if and only if p.d.$Q\leq 1$ (see e.g. [15, Chap. VII, Theorem
2.8]).

(ii) See [15, Chap. VII, Sect. 2]. ∎

Injective objects. The role of injective modules is played by objects
$E\in{\mathcal{C}}_{V}^{*}$ that satisfy $\mathop{\rm
Ext}\nolimits_{V}^{1}(A,E)=0$ for all $A\in{\mathcal{C}}_{V}^{*}$; i.e.
whenever $E$ is a subobject, it must be a summand. Luckily, this property is
equivalent to the more familiar extensibility of morphisms into $E$ from
subobjects to objects. But this equivalence comes with a caveat: the extended
map need not be a ${\mathcal{C}}_{V}^{*}$-morphism, since its image might not
be tight. Perhaps unexpectedly, injectivity and divisibility turn out to be
equivalent.

######  Theorem 6.6.

The following conditions are equivalent for an object
$E\in{\mathcal{C}}_{V}^{*}$.

(i) $\mathop{\rm Ext}\nolimits_{V}^{1}(C,E)=0$ for all
$C\in{\mathcal{C}}_{V}^{*}$;

(ii) every morphism ${\phi}:A\to E$ in ${\mathcal{C}}_{V}^{*}$ extends to a
homomorphism $\psi:B\to E$ whenever $A,B$ are in ${\mathcal{C}}_{V}^{*}$ and
$A$ is tight in $B$;

(iii) $E$ is a divisible object.

###### Proof.

In the category ${\mathcal{C}}_{V}$ we have an exact sequence

$None$ $\mathop{\rm Hom}\nolimits_{V}(B,E)\to\mathop{\rm
Hom}\nolimits_{V}(A,E)\to\mathop{\rm Ext}\nolimits_{V}^{1}(B/A,E)\to\dots$

(i) $\Rightarrow$ (ii). Hypothesis implies that Ext in (4) vanishes, so the
map between the two Homs is surjective.

(ii) $\Rightarrow$ (iii). Condition (ii) ensures that for every $r\in
V^{\times}$, every map ${\phi}:Vr\to E$ extends to $V\to E$ (note that
${\phi}\in{\mathcal{C}}_{V}^{*}$). This is equivalent to the divisibility of
$E$ by $r$.

(iii) $\Rightarrow$ (i). By Bazzoni–Herbera [3], over an integral domain $R$,
a module $E$ is divisible if (and only if) $\mathop{\rm
Ext}\nolimits_{R}^{1}(C,E)=0$ holds for all $R$-modules $C$ of p.d.$\leq 1$.
(This implication was proved in [9, Theorem 6] for Prüfer domains.) ∎

As an immediate corollary to the preceding theorem we obtain:

######  Corollary 6.7.

Direct sums of injective objects are likewise injective. In particular,
injective objects are $\Sigma$-injective, i.e. any direct sum of copies of an
injective object is injective. ∎

A well-known test for the injectivity of a module is that its extensions by
cyclic modules are splitting. In ${\mathcal{C}}_{V}^{*}$ this criterion
simplifies to cyclically presented modules:

######  Theorem 6.8.

An object $E\in{\mathcal{C}}_{V}^{*}$ is injective if and only if $\mathop{\rm
Ext}\nolimits_{V}^{1}(C,E)=0$ holds for all cyclically presented objects $C$.
∎

It is well known in commutative module theory that every module contains a
unique maximal divisible submodule. This is not true in
${\mathcal{C}}_{V}^{*}$ in general, because the relevant property that the sum
of two divisible objects is again one fails; indeed, the property of being of
p.d. at most $1$ is frequently lost when forming the sum.

######  Example 6.9.

We exhibit an injective object which is a mixed $V$-module, but neither
torsion, nor torsion-free, nor mixed as an object of ${\mathcal{C}}_{V}^{*}$.
Let $V$ be a valuation domain such that p.d.$Q=3$, and consider the divisible
$V$-module $\partial_{V}$ defined above. As p.d.$\partial_{V}=1$, we have
$\partial_{V}\in{\mathcal{C}}_{V}^{*}$. The torsion submodule $T$ of
$\partial_{V}$ has p.d.2, since $\partial_{V}/T\cong Q$. Therefore
$\partial_{V}$, as an object of ${\mathcal{C}}_{V}^{*}$, is neither torsion,
nor torsion-free, nor mixed. ∎

The following corollary is obvious in view of our discussion of the divisible
objects. (For the following (i), cf. Theorem 6.4.)

######  Corollary 6.10.

(i) Every object embeds as a subobject in an injective object.

(ii) Objects that are epic images of injective objects $($modulo subobjects$)$
are themselves injective.

(iii) Every object $M$ admits an injective resolution, that is an exact
sequence $0\to M\to A\to B\to 0$ of modules of p.d.$\leq 1$ where $A,B$ are
injective objects.

(iv) The injective dimension of any object in ${\mathcal{C}}_{V}^{*}$ is $0$
or $1$. ∎

Let us pause for a moment to answer a question concerning the existence of
injective envelopes in the class ${\mathcal{C}}_{V}^{*}$. By the injective
envelope of an object $M$ we mean an injective object $E(M)$ containing $M$ as
a subobject such that for every injective object $E$ containing $M$ as a
subobject, the identity map of $M$ extends to a tight embedding $E(M)\to E$.
Of course, if an envelope exists, it is then unique up to isomorphism.

######  Theorem 6.11.

All modules in the class ${\mathcal{C}}_{V}^{*}$ admit injective
$($divisible$)$ envelopes if and only if $V$ is a rank one discrete valuation
domain.

###### Proof.

If $V$ is a DVD, then ${\mathcal{C}}_{V}$ and ${\mathcal{C}}_{V}^{*}$ are
identical, and the claim in ${\mathcal{C}}_{V}$ is well-known. If $V$ is a
Matlis domain, then all divisible modules are $h$-divisible, they are direct
sums of copies of $Q$ and/or $K$. If $V$ is not a DVD, then it contains a
countably generated ideal, and such an ideal cannot be tight in a direct sum
of $Q$s. Finally, if p.d.$Q>1$, then the ${\mathcal{C}}_{V}^{*}$-injective
envelope of $V$ should be an indecomposable summand of $\partial_{V}$, but —
as observed above — such a summand does not exist. ∎

##  7\.  Pure-Injectivity

Pure-injectivity in ${\mathcal{C}}_{V}^{*}$ can be developed like in
traditional module theory (see e.g. [14, Chap. XI, Section 2] and [15, Chap.
XIII, Sections 2-3]). As expected, there are several changes, so we provide
the whole proof, but skip routine arguments.

By a system of equations (with unkowns $x_{j}\ (j\in J)$) over a module $M$ we
mean a system of linear equations

$None$ $\sum_{j\in J_{i}}r_{ij}x_{j}=a_{i}\in M\qquad(r_{ij}\in V,\ i\in I)$

for $i\in I,\ j\in J$ where $I,J$ are arbitrary index sets, and $J_{i}$ is a
finite subset of $J$ for each $i\in I$. Let $F$ denote the free module
generated by the unknowns $x_{j}\ (j\in J)$ and $H$ its submodule generated by
the left sides of the equations for all $i\in I$. We consider only consistent
systems; i.e., systems that do not contain hidden contradiction. This means
that we get a genuine homomorphism ${\phi}:H\to M$ by mapping the generators
of $H$ onto the elements of $M$ as shown by the equations, i.e.
${\phi}:\sum_{j\in J_{i}}r_{ij}x_{j}\mapsto a_{i}$. It is easy to check that
the system has a solution in $M$ if and only if ${\phi}$ extends to a
homomorphism ${\phi}^{*}:F\to M$, in which case $x_{i}={\phi}^{*}(x_{i})\in M$
are the solutions. It is pretty obvious that a consistent system defines an
extension $M^{*}$ of $M$ by $F/H$ by adjoining to $M$ the unknowns $x_{i}$ as
generators subject to the defining relations (5). Furthermore, it is
straightforward to check that $M$ will be pure in $M^{*}$ exactly if (5) is
finitely solvable in $M$, i.e. the finite subsystems of (5) admit solutions in
$M$.

We define p.d.($F/H$) as the projective dimension of the system (5). It will
be convenient to call (5) an adequate equation system if 1) it is consistent;
2) its p.d. is $\leq 1$; and 3) it is finitely solvable.

An object $M\in{\mathcal{C}}_{V}^{*}$ is said to be pure-injective if it has
either one of the equivalent properties listed in the following theorem.

######  Theorem 7.1.

For an object $M$, $({\alpha})$-$({\gamma})$ are equivalent properties:

$({\alpha})$ $\mathop{\rm Pext}\nolimits_{V}^{1}(C,M)=0$ for all objects $C$.

$({\beta})$ If $A$ is a pure subobject of object $B$, then every
${\mathcal{C}}_{V}^{*}$-map $A\to M$ extends to a homomorphism $B\to M$
$($that need not be a ${\mathcal{C}}_{V}^{*}$-map$)$.

$({\gamma})$ Every adequate equation system over $M$ has a global solution in
$M$.

###### Proof.

All modules in this proof are objects in ${\mathcal{C}}_{V}^{*}$.

(${\alpha})\Rightarrow({\beta}$) Assuming $({\alpha})$, consider the following
push-out diagram where the top sequence is pure-exact and $\zeta$ is a
${\mathcal{C}}_{V}^{*}$-morphism.

$\begin{CD}0@>{}>{}>A@>{}>{}>B@>{}>{}>C@>{}>{}>0\\\
&&@V{}V{\zeta}V@V{}V{}V\Big{\|}\\\ 0@>{}>{}>M@>{}>{}>N@>{}>{}>C@>{}>{}>0\\\
\end{CD}$

Then the bottom row is also pure-exact, so it splits by hypothesis. Hence
there is a homomorphism $B\to M$ making the upper triangle $ABM$ commute. This
is an extension of $\zeta$, proving $({\beta})$.

(${\beta})\Rightarrow({\gamma}$) Given an adequate equation system (5) over
$M$, consider the corresponding free module $F$ and its submodule $H$. If (5)
is viewed as a system over $M^{*}$, then by construction, there is an
extension $\psi:F\to M^{*}$ of ${\phi}:H\to M\leq M^{*}$. This means that (5)
is solvable in $M^{*}$. Hypothesis $({\beta})$ implies that $M$ is a summand
of $M^{*}$. Hence $\psi$ followed by the projection $M^{*}\to M$ yields a
desired extension $F\to M$ of ${\phi}$.

(${\gamma})\Rightarrow({\alpha}$) In the next diagram, let the bottom row
represent a pure extension of $M$ by $C$, and the top row a free resolution of
$C$. The map ${\phi}^{*}$ exists because $F$ is free (making the right square
commute). It is evident that its restriction ${\phi}$ to $H$ makes the left
square commute. As $H$ is tight in $F$, the pair $\\{H,F\\}$ along with
${\phi}$ defines a system (5) of equations that is finitely solvable in $M$,
due the purity of the bottom sequence. Thus (5) is an adequate system, and
hence condition $({\gamma})$ implies that there exists a map $F\to M$ that
makes the maps in the upper triangle $HFM$ commute. Then the bottom sequence
splits, establishing (${\alpha}$).

$\begin{CD}0@>{}>{}>H@>{}>{}>F@>{}>{}>C@>{}>{}>0\\\
&&@V{{\phi}}V{}V@V{}V{{\phi}^{*}}V\Big{\|}\\\
0@>{}>{}>M@>{}>{}>N@>{}>{}>C@>{}>{}>0\\\ \end{CD}$

∎

Evidently, divisible (i.e. injective) objects are pure-injective. Moreover,
they contain a lot of pure-injective subobjects — as is shown by the following
theorem.

######  Theorem 7.2.

Let $D$ be a divisible object in ${\mathcal{C}}_{V}^{*}$. For every $r\in
V^{\times}$, the submodule $D[r]=\\{d\in D\ |\ rd=0\\}$ is a pure-injective
object.

###### Proof.

From the isomorphism $D/D[r]\cong D$ and from p.d.$D=1$ we infer that $D[r]$
is tight in $D$. Let $A$ be a pure subobject of object $B$, and $\xi:A\to
D[r]$ a ${\mathcal{C}}_{V}^{*}$-map. $\xi$ induces a map $\xi^{\prime}:A/rA\to
D[r]$ that extends (by purity) to $\xi^{\prime\prime}:B/rB\to D$. Evidently,
$\mathop{\rm Im}\xi^{\prime\prime}\leq D[r]$ as well, so the canonical map
$B\to B/rB$ followed by $\xi^{\prime\prime}$ yields a desired extension $B\to
D[r]$ of $\xi$. ∎

Imitating the proof of Eklof–Mekler [5, Chap. V, Corollary 1.3], we verify:

######  Theorem 7.3.

Every object in ${\mathcal{C}}_{V}^{*}$ is a pure subobject in a pure-
injective object.

###### Proof.

Select any cardinal ${\kappa}>\max\\{|V|,{\aleph}_{0}\\}$. Given a module
$M\in{\mathcal{C}}_{V}^{*}$, we define a continuous well-ordered ascending
chain $\\{M_{\sigma}\ |\ {\sigma}<{\kappa}\\}$ of length ${\kappa}$ as
follows. Start with $M_{0}=M$. If for some ${\sigma}$ the modules $M_{\rho}$
of p.d.$\leq 1$ have been constructed for all ${\rho}\leq{\sigma}$, then
define $M_{{\sigma}+1}$ by adjoining to $M_{\sigma}$ the unknowns (as
additional generators) of every adequate equation system with defining
relations given by the systems. It is readily checked that then
p.d.$M_{{\sigma}+1}\leq 1$ as well, and $M_{\sigma}$ will be tight and pure in
$M_{{\sigma}+1}$ such that all the adequate equation systems over $M_{\sigma}$
are solvable in $M_{{\sigma}+1}$. At limit ordinals, we take the union which
will again have p.d.$\leq 1$ and will contain all previously constructed
$M_{\rho}$s as tight pure submodules. It is straightforward to check that the
union of the constructed chain will satisfy condition $({\gamma})$, and thus
it will be a pure-injective object containing $M$ as a pure subobject. (The
process can stop at systems with ${\lambda}$ unknowns, where ${\lambda}$ is
any uncountable cardinal $>|V|.)$ ∎

The next proposition implies that the first Ulm-submodule of a pure-injective
object is an injective object whenever its p.d. is $\leq 1$.

######  Proposition 7.4.

The first Ulm-submodule $M^{1}=\cap_{r\in V^{\times}}rM$ of a pure-injective
object $M$ satisfies $\mathop{\rm Ext}\nolimits_{V}^{1}(C,M^{1})=0$ for all
objects $C$.

###### Proof.

Let $\\{r_{i}\ |\ i\in I\\}$ be a list of the non-unit elements of
$V^{\times}$. We have to show that for any given $a\in M^{1}$, for each
$r_{j}$ the equation $r_{j}x=a$ is solvable for $x\in M^{1}$. For each $j\in
I$, consider the following system of linear equations

$r_{j}x=a,\quad r_{i}x_{i}=x\quad(i\in I).$

An easy calculation confirms that the p.d. of this system is $1$. Furthermore,
since the chosen element $a$ is divisible by $r_{j}r_{i}$ for all indices,
each system is finitely solvable in $M.$ By hypothesis, it has a solution in
$M$. Clearly, each solution $x$ belongs to $M^{1}$, so $M^{1}$ is a divisible
submodule. ∎

For every valuation domain $V$ of global dimension $\geq 2$, we exhibit an
example of a pure-injective object in ${\mathcal{C}}_{V}^{*}$ whose injective
submodule is an object, but neither a subobject nor a summand in
${\mathcal{C}}_{V}^{*}$.

######  Example 7.5.

Let $V$ be as stated. We form the following diagram with pure-exact rows and
commutative squares.

$\begin{CD}0@>{}>{}>A@>{}>{}>B@>{}>{}>C@>{}>{}>0\\\
&&@V{}V{}V@V{}V{}V\Big{\|}\\\
0@>{}>{}>D@>{}>{}>B^{\prime}@>{}>{}>C@>{}>{}>0\\\
&&\Big{\|}@V{}V{}V@V{}V{}V\\\
0@>{}>{}>D@>{}>{}>B^{\prime\prime}@>{}>{}>C^{\prime}@>{}>{}>0\\\ \end{CD}$

For the top row select a pure-exact sequence of torsion $V$-modules such that
$A,B$ are of p.d.1, and p.d.$C=2$. We can make the selection such that the
first Ulm-submodule of $C$ is $0$. Embed $A$ in a divisible object $D$ as a
tight submodule, and get the middle row as a pure-exact sequence via pushout.
The next step is the application of the embedding process of Theorem 7.3 to
$C$ (it works even if $C$ is not an object) to obtain a pure extension
$C^{\prime}$ by a module $H$ of p.d.$1$ such that $C^{\prime}$ has the
property that its pure extensions by $V$-modules of p.d.$\leq 1$ are
splitting. Since p.d.$H=1$, there is a module $B^{\prime\prime}$ making
(though not uniquely) the bottom sequence pure-exact and the diagram
commutative. The middle vertical arrows are injections and the projective
dimensions of $B,B^{\prime}/B,B^{\prime\prime}/B^{\prime}$ are all 1. Hence we
have p.d.$B^{\prime\prime}=1$ as well. Furthermore, as a pure extension of $D$
by $C^{\prime}$, the module $B^{\prime\prime}$ has the property that its pure
extensions by $V$-modules of p.d.$\leq 1$ are splitting. This means that
$B^{\prime\prime}$ is a pure-injective object in ${\mathcal{C}}_{V}^{*}$. Its
injective submodule $D$ has p.d.1, but, since p.d.$C^{\prime}=2$, it is not
tight in $B^{\prime\prime}$, so it is not a summand in
${\mathcal{C}}_{V}^{*}$. ∎

We close this section with an example and its corollary.

######  Example 7.6.

An explicit example of a pure-injective object is a direct sum $C=\oplus_{i\in
I}V/Vr$ for any non-unit $r\in V^{\times}$ and any index set $I$. This follows
from Proposition 5.4 (iii). ∎

Consequently, we can state the following corollary (the case for finitely
presented objects has been stated above in Theorem 5.1):

######  Corollary 7.7.

Every $V/Vr$-homogeneous $(r\in V^{\times})$ torsion module is $\Sigma$-pure-
injective.

###### Proof.

If $D$ is any direct sum of copies of $\partial_{V}$, then the submodule
$D[r]$ is $V/Vr$-homogeneous, so a direct sum of copies of $V/Vr$. Moreover,
it is pure-injective by Example 7.6. Summands of $D[r]$ as well as finite
direct sums of pure-injectives are also pure-injective. ∎

##  8\.  Cotorsion Modules

We shall call an object $C$ cotorsion if $\mathop{\rm
Ext}\nolimits^{1}_{R}(U,C)=0$ holds for all uniserial (i.e. rank one) torsion-
free objects $U$. Evidently, it suffices to demand this only for countably
generated uniserials. Readers familiar with cotorsion theory immediately
recognize that this cotorsion concept corresponds to Warfield-cotorsion (where
splitting is required for extensions by all torsion-free modules). This claim
will become even more transparent in light of the following general statement.

######  Lemma 8.1.

An object $C\in{\mathcal{C}}_{V}^{*}$ is cotorsion if and only if it satisfies
the equation $\mathop{\rm Ext}\nolimits^{1}_{V}(A,C)=0$ for all torsion-free
objects $A$.

###### Proof.

Definition settles the claim in one direction. For the ’only if’ part, assume
that $C$ is cotorsion and $A$ is torsion-free of p.d.$\leq 1$. We know from
Theorem 5.10 that then $A$ is the union of a continuous well-ordered ascending
chain of torsion-free submodules $A_{\alpha}\ ({\alpha}<{\kappa})$ that are
pure and tight in $A$ such that all the quotients $A_{{\alpha}+1}/A_{\alpha}$
are torsion-free of rank 1. We now refer to Eklof’s theorem (see [4]) on the
extension by the union of a chain to conclude that $A$ satisfies the quoted
equation, as all the mentioned quotients in the chain satisfy it. ∎

In order to characterize cotorsion objects in terms of solvability of systems
of equations, consider a torsion-free uniserial object $U$. If it is not
cyclic, then it is generated by a countable set $\\{u_{n}\ |\ n<{\omega}\\}$
such that $r_{n}u_{n+1}=u_{n}$ for some $r_{n}\in V^{\times}$ for all
$n<{\omega}$. Therefore, an extension $B$ of module $C$ by $U$ looks like
$B=\langle C,b_{n}\ (n<{\omega})\rangle$ with defining relations given by the
equations $r_{n}b_{n+1}-b_{n}=c_{n}$ for certain $c_{n}\in C$. Clearly, $C$ is
cotorsion if and only if $C$ is a summand of $B$ for all permissible choices
of the $U$s and the $c_{n}$s if and only if each consistent countable system
of equations of the form

$None$ $r_{n}x_{n+1}-x_{n}=c_{n}\quad{\rm with}\ c_{n}\in C\ (n<{\omega})$

is solvable in $C$. In the last case, a solution $x_{n}$ yields a complement
to $C$ in $B$: the submodule generated by the elements $a_{n}=x_{n}-b_{n}\
(n<{\omega})$. This leads us to the following theorem.

######  Theorem 8.2.

An object $C$ is cotorsion if and only if all consistent systems of linear
equations of the form $(6)$ constructed with torsion-free uniserial objects
$U$ are solvable in $C$. ∎

More useful information about cotorsion objects is provided by the next
result.

######  Theorem 8.3.

(i) All pure-injective objects in ${\mathcal{C}}_{V}^{*}$ are also cotorsion
objects.

(ii) Every object is a subobject of a cotorsion object with torsion-free
cokernel.

###### Proof.

(i) is an immediate consequence of the definitions, since all extensions by
torsion-free $V$-modules are pure-extensions.

(ii) This can be verified easily, just copy the proof of Theorem 7.3, using
(6) in place of linear systems of p.d.$\leq 1$, mutatis mutandis. (By the way,
the mere embedding property follows already from (i) and Theorem 7.3.) ∎

The next lemma is a convincing evidence that in some respect the cotorsion
objects behave like ordinary cotorsion modules, though several relevant
features are missing.

######  Lemma 8.4.

(i) Extension of cotorsion by cotorsion is again cotorsion.

(ii) Modules of p.d.$1$ that are epimorphic images of cotorsion objects
$($modulo tight submodules$)$ are likewise cotorsion objects.

###### Proof.

(i) is obvious.

(ii) This is evident considering that if $C\to C^{\prime}$ is a surjective
map, then for every module $A$ of p.d.$\leq 1$, the induced map $\mathop{\rm
Ext}\nolimits^{1}_{V}(A,C)\to\mathop{\rm Ext}\nolimits^{1}_{V}(A,C^{\prime})$
is also surjective. ∎

In order to demonstrate that not all cotorsion objects are pure-injective,
take e.g. a torsion object $T$ whose first Ulm-submodule $T^{1}$ is a
subobject, but not divisible. Then the embedding process mentioned in the
proof of Theorem 8.3(ii) yields a cotorsion object $\overline{T}$ containing
$T$ as a tight submodule such that $\overline{T}/T$ is torsion-free. This
$\overline{T}$ cannot be not pure-injective, because its Ulm-submodule
contains its torsion submodule $T^{1}$ that is a subobject, but is not
injective (cf. Theorem 7.4). (Another proof can be given by displaying an
extension of a pure-injective by a pure-injective (that is necessarily
cotorsion) which fails to be pure-injective.)

We raise the following problem on cotorsion modules in
${\mathcal{C}}_{V}^{*}$.

######  Problem 8.5.

Are the Ulm-submodules of cotorsion modules cotorsion and the Ulm-factors
pure-injective in ${\mathcal{C}}_{V}^{*}$ as in the case of DVD? ∎

Acknowledgment. We would like to thank Luigi Salce for his numerous helpful
comments.

Correction. (by L. Fuchs) Non-standard uniserial modules appear frequently in
the study of modules over valuation domains, we could not avoid mentioning
them in our study either. I would like to correct erroneous statements on them
in the literature. In her very interesting papers on non-standard uniserials
(Bull. Amer. Math. Soc. 25 (1991) and Contemporary Math. 124 (1992)) B.
Osofsky stated that non-standard uniserials were investigated because of their
connection to Kaplansky’s problem on the existence of valuation rings that are
not homomorphic images of valuation domains. This incorrect claim (with
another mistaken statement) was restated in the review of Osofsky’s first
article by R. Göbel in Math. Reviews. The fact is that the problem of
existence of non-standard uniserials was raised in 1980 by L. Salce during our
joint investigation of modules over valuation domains, and it was him who
named them ”non-standard”. S. Shelah was told about non-standard uniserials
only at the Udine Conference in April 1984 just before the night he succeeded
in establishing their existence. Then neither Shelah nor anybody else at the
well-attended conference could claim that Kaplansky’s problem had been solved,
since at this point nobody suspected that it was related to non-standard
uniserials. The connection became known only three months later when we solved
the Kaplansky problem, and the solution relied on non-standard uniserials (see
the original solution published in [14]).

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