# A Universal Property of the Spectrum of a Ring and the Semiring of its
Ideals

William Bernardoni
Case Western Reserve University
Cleveland, OH 44118
<EMAIL_ADDRESS>
###### Abstract

Arithmetic valuations are intimately connected with the structure of the
ideals of a commutative ring. We show how the generalized idempotent semiring
valuations of Jeffrey and Noah Giansiracusa can be used to make this
connection explicit. Through this generalized valuation theory sufficiently
complete positive valuations give rise to Galois correspondences with the
lattice of ideals of a commutative ring. Up to isomorphism the semiring of
ideals of a commutative ring can be defined as a universal factoring semiring
for positive valuations. We then further show that this valuation theory can
formally connect Joyal’s notion de zéros and universal property of the
spectrum of a ring to arithmetic valuation theory. We show that up to
isomorphism as a coframe, the closed Zariski topology on a spectrum of a
commutative ring can be defined as the universal factoring semiring of
additively and multiplicatively idempotent valuations.

## Disclosure Statement

The author reports there are no relevant financial or non-financial competing
interests in the publication of this research.

##  1 Introduction

Arithmetic valuations are a powerful tool connecting algebraic geometry,
tropical geometry, and non-Archimedean analysis. They were an integral
component of Zariski’s conception of algebraic geometry, and are intimately
connected with $p$-adic numbers and $p$-adic analysis. Equipping a ring with a
valuation allows one to tropicalize, bringing in additional tools to study
algebro-geometric objects over the ring, such as the method of patchworking
created by Oleg Viro to classify isotropy classes of real algebraic curves of
degree 7 [Vir06, IMS07].

Saying that a ring admits a non-degenerate valuation is equivalent to placing
a non-Archimedean absolute value on the ring. There are many functionals on
algebras however that appear similar to standard valuations but are not
exactly non-Archimedean absolute values. This insight led to generalizations
of arithmetic valuations.

One such generalized notion is due to Krull and such valuations are called
Krull valuations, wherein the codomain of the valuation was taken to be a
totally ordered abelian semigroup rather than strictly the extended real
numbers. Zariski used and extended this notion to form a basis for his
algebraic geometry.

In their paper on a scheme theoretic tropicalization of toric varieties,
[GG16], Noah and Jeffrey Giansiracusa introduced a further generalization of a
valuation, where the codomain was taken to be an idempotent semiring. This
allows one to consider partially ordered valuations and extend the idea of
tropicalization to further semirings. This approach can also be used to define
a non-commutative valuation theory, which was partially explored in [Ber23].
This generalization allows one to connect the standard theory of arithmetic
valuations with similar notions, such as Big-O asymptotic complexity in
computer science, and assigning zero sets to polynomials.

The valuations on a ring are strongly connected with the underyling structures
of the ring. Recent work by Noah and Jeffrey Giansiracusa have shown that in a
scheme-theoretic setting their generalized notion of a valuation is intimately
connected with the Berkovich analytification [GG22]. Classical results, such
as Krull’s classification of valuation rings [Kru39], show that the existence
of standard arithmetic valuations is intimately tied to the structure of the
lattice of ideals of a given ring.

In this paper we will clarify the connection between the structure of ideals
of a commutative ring and the valuations of that ring. We will also show how
the generalized valuations that exist on a ring determine and are determined
by its prime spectrum. To do so we explore two further specializations of the
Giansiracusas’ valuation theory and use them to derive defining universal
properties for the semiring of ideals of a commutative ring as well as the
coframe of the closed Zariski topology on its prime spectrum.

We will first examine positive valuations – valuations whose codomain are
idempotent semirings bounded by $0$ and $1$, and show that the semiring of
ideals of a commutative Noetherian ring is the universal semiring that factors
all such positive valuations. When we further restrict the codomain to
complete idempotent semirings bounded by $0$ and $1$, also known as
commutative integral quantales, and show that for any commutative ring its
quantale of ideals is defined up to isomorphism as the universal quantale that
factors such maps.

As a result we have in Theorem 4.15 that every sufficiently complete positive
valuation induces a Galois connection on the ideals of a commutative ring and
Corollary 4.21 stating that the semiring of ideals may be defined up to
isomorphism by examining the positive valuations of a commutative ring.

Theorem 4.15. Let $R$ be a commutative ring and $I(R)$ its semiring of ideals.

Let $\nu:R\to\Gamma$ be a positive valuation, and let $\Gamma$ be a complete
semiring. Then there exists a Galois connection between $I(R)$ and $\Gamma$.

That is, there exist monotone maps $N:I(R)\to\Gamma$ and $I:\Gamma\to I(R)$
such that:

$\gamma\leq N(J)\iff J\subseteq I(\gamma)$

For $\gamma\in\Gamma$, $J\in I(R)$

Corollary 4.21. Let $R$ be a commutative ring.

$I(R)$ is initial among positive valuations whose valuation semiring is
complete for sums of size $|R|$ or less.

If $R$ is Noetherian then $I(R)$ is initial among all positive valuations.

We will then restrict further to multiplicatively idempotent positive
valuation – valuations whose codomain are appropriately bounded semirings that
are multiplicatively and additively idempotent. In the Noetherian case the
Zariski closed topology on the prime spectrum is the universal semiring that
factors such maps. If we restrict to the complete case, and study valuations
whose codomain is a coframe (a complete, multiplicative and additively
idempotent semiring), then for any commutative ring the Zariski closed
topology is the universal coframe that factors all such valuations.

This will culminate in Theorem 5.11.

Theorem 5.11. Let $R$ be a commutative ring.

The initial valuation among appropriately complete, multiplicatively
idempotent positive valuations is $\tau:R\to T$, where $T$ is the coframe of
the closed Zariski topology on the spectrum of a commutative ring and
$\tau(r)$ is the collection of prime ideals containing the element $r$.

One of the core results of this paper, that the spectrum of a ring is the
“universal zero locus", is a folkloric result. This result originated in the
note by André Joyal, [Joy76], where it was stated that the prime spectrum of a
ring factors all notion de zéros – valuations whose image is in a distributive
lattice. This result also appears in the book, [Joh82] in section V.3.1 where
it is used as an equivalent definition of the Zariski spectrum of a ring. This
result is mentioned and utilized in a Mathematics Stack Exchange answer by
Zhen Lin Low, [Low]. While this particular result may not be novel, there is
novelty in our approach, and value in explicitly stating this result and proof
in terms of valuations of commutative rings. This work fully connects Joyal’s
notion de zéros with classical arithmetic valuation theory.

While this work is focused on the valuation theory of commutative rings, there
is recent work developing a valuation theory applied to idempotent semirings
themselves. We can find similar results in this other algebraic context. In
the work [Tol16], valuations are taken to be any homomorphism between an
idempotent semiring and a totally ordered one. The central result in [Ber23]
can be seen as a ring-theoretic analog of a central result in [Tol16]. In
[Tol16], Jeffrey Tolliver finds that the universal valuation on an idempotent
semiring is given by the collection of $\mathbb{Z}$-submodules, we see this
reflected in the ring case in [Ber23] where the universal valuation on a ring
is given by the collection of $\mathbb{Z}$-submodules of the ring. The work
[JRT21] finds that the space of surjective valuations between idempotent
semirings is isomorphic to the prime spectrum of that semiring.

## Outline

  * •

In Section 2 we introduce the algebraic definitions we will use. Section 2.1
gives definitions and examples of semirings, idempotent semirings, and
complete idempotent semirings. A reader familiar with semirings may skip this
section, noting that we use the order convention that $a\leq b\iff a+b=a$.
Section 2.2 gives the generalized definition of an idempotent semiring
valuation in the sense of [GG16].

  * •

In Section 3 we give the definition of a universal valuation (Definition 3.3),
and repeat prior results from [Ber23] giving a classifying structural theorem
(Theorem 3.5) to these universal valuation semirings. We give an explicit
example (Example 3.3) of a non-trivial, non-commutative universal valuation
semiring.

  * •

In Section 4 we introduce our notion of a positive valuation (Definition 4.1),
we show that every positive valuation gives rise to a Galois connection with
the lattice of ideals of a commutative ring (Theorem 4.15) and we show that
with minimal assumptions the universal positive valuation semiring of a
commutative ring $R$ is the semiring of ideals of that ring $I(R)$ (Corollary
4.21). We then show how one can recover many properties of the ideals of a
ring purely from the semiring structure of $I(R)$.

  * •

In Section 5 we restrict further to multiplicatively idempotent positive
valuations (Definition 5.2) – e.g. coframe or set valued valuations – and we
show how the generalized notion of a valuation given by the Giansiracusas can
be used to link Joyal’s notion de zéros, [Joy76] and Johnstone’s defining
universal property of the Zariski spectrum (Section V.3.1 of [Joh82]) of a
ring to classical arithemetic valuation theory. To do so we show that the
universal multiplicatively idempotent positive valuation semiring of a
commutative ring is given by the closed Zariski topology on its prime spectrum
(Theorem 5.11).

## Acknowledgments

We would like to thank Nick Gurski for his mentorship, conversations and
advice that helped shape this paper. We would also like to thank Zhen Lin Low
for his correspondence and help in finding the references to Joyal and
Johnstone’s work. We would also like to thank Roland Baumann, Stephanie Egler,
and Andrew Edwards for great mathematical conversations and feedback.

##  2 Algebraic Preliminaries

This section is adapted from section 2 of [Ber23].

Conventions:

  * •

In this paper we assume semirings, rings, and monoids are unital.

  * •

We denote the powerset of a set $X$ as $2^{X}$.

  * •

In equations that involve multiple algebraic objects, we denote by $+_{S}$ and
$*_{S}$ to be the addition and multiplication respectively in the object $S$.
In particular when we refer to standard arithmetic we will note
$+_{\mathbb{R}}$ and $*_{\mathbb{R}}$ to refer to the standard addition and
multiplication over the real numbers.

  * •

In general we will use $R$ to designate an arbitrary ring, $S$ an arbitrary
semiring, and $\Gamma$ an arbitrary idempotent semiring.

###  2.1 Semirings

In this paper we will be concerned with valuations whose codomain are
idempotent semirings.

######  Definition 2.1.

A semiring $(S,+,*,0_{S},1_{S})$ is a tuple where $S$ is a set, $+$ and $*$
are binary operations on $S$, and $0_{S}$ and $1_{S}$ are elements of $S$ such
that:

  1. 1.

$(S,+)$ is a commutative monoid with identity element $0_{S}$.

  2. 2.

$(S,*)$ is a monoid with identity element $1_{S}$.

  3. 3.

Multiplication distributes over addition.

  4. 4.

Multiplication by $0_{S}$ annihilates $R$, that is $0_{S}*s=0_{S}=s*0_{S}$ for
all $s\in S$.

Remark 2.2. A ring is a semiring whose additive monoid is a group and not just
a monoid. Semirings can be seen as “rings without negation", and so they are
sometimes called rigs.

The fact that multiplication by $0$ annihilates in a ring can be proven from
the group structure of addition. In a semiring we cannot subtract, and so this
must be taken as an axiom.

######  Definition 2.3.

A semiring $S$ is called idempotent if for all $a\in S$: $a+a=a$.

The additive idempotency of a semiring is entirely determined by the
idempotency of its multiplicative unit.

######  Proposition 2.4.

A semiring $S$ is idempotent if and only if $1_{S}+1_{S}=1_{S}$.

######  Definition 2.5.

We refer to the category of idempotent semirings as $\mathbf{ISR}$.

An object in $\mathbf{ISR}$ is an idempotent semiring, and a morphism
$\phi:S\to{S^{\prime}}$ is a unital semiring homomorphism. That is for all
$a,b\in S$:

  * •

$\phi(a+_{S}b)=\phi(a)+_{S^{\prime}}\phi(b),$

  * •

$\phi(a*_{S}b)=\phi(a)*_{S^{\prime}}\phi(b),$

  * •

$\phi(0_{S})=0_{S^{\prime}},$

  * •

$\phi(1_{S})=1_{S^{\prime}}.$

######  Definition 2.6.

A semiring $S$ is called commutative if for all $a,b\in S$: $ab=ba$.

We will chiefly be concerned with commutative idempotent semirings in this
paper.

Example 2.7.  The most famous example of a commutative idempotent semiring is
the tropical semiring

$\mathbb{T}=(\mathbb{R}\cup\\{\infty\\},\min,+_{\mathbb{R}},\infty,0).$

The tropical numbers are the extended real numbers with addition being the
minimum of two numbers and multiplication being standard addition.

$a+_{\mathbb{T}}b=\min(a,b),$ $a*_{\mathbb{T}}b=a+_{\mathbb{R}}b.$

The tropical semiring is an important structure in computer science as well as
in algebraic geometry. Some good introductions to the applications of the
tropical semiring and its geometry are the books [MS15, IMS07, HOW06].

Remark 2.8. There are three equivalent formulations of the tropical semiring.
The one we use in this paper is the ($\min$,+) tropical semiring as it is the
formulation that appears in the standard definition of an arithmetic
valuation.

The other two forms are the $(\max,+)$ tropical semiring:
$(\mathbb{R}\cup\\{-\infty\\},\max,+_{\mathbb{R}},-\infty,0)$ and the
$(\max,\times)$ tropical semiring:
$([0,\infty),\max,\times_{\mathbb{R}},0,1)$.

The $(\max,+)$ and the $(\min,+)$ semirings are isomorphic via the map

$x\mapsto-x.$

The $(\max,+)$ and the $(\max,\times)$ semiring are isomorphic via the map

$x\mapsto e^{x}.$

Example 2.9.  Another core example of a commutative idempotent semiring is the
boolean semiring

$\mathbb{B}=(\\{\bot,\top\\},\vee,\wedge,\bot,\top)$

where $\bot$ represents the logical false, $\top$ the logical true, $\vee$ is
logical OR and $\wedge$ is logical AND.

Remark 2.10. The boolean semiring is the unique idempotent semiring of order
$2$ and is the initial object in $\mathbf{ISR}$.

Remark 2.11. All semirings can be seen as $\mathbb{N}$ algebras. Rings are
exactly the semirings that have a $\mathbb{Z}$ algebra structure, and
idempotent semirings are exactly the semirings that admit a $\mathbb{B}$
algebra structure.

######  Definition 2.12.

Idempotent semirings carry a natural partial order, where we define

$a\leq b\iff a+b=a.$

We say $a<b$ if $a\leq b$ and $a\neq b$.

This is in analogy with the ordering on the tropical semiring,
$\mathbb{T}=(\mathbb{R}\cup\\{\infty\\},\min,+,\infty,0)$. A useful trick to
remember this convention is that “the baker eats the bigger baguette," as, if
$b\geq a$, then $b$ “is eaten" in the sum. Some authors prefer the flipped
convention where $a\geq b$ if and only if $a+b=a$.

Example 2.13.  In $\mathbb{B}$ we have that $\top<\bot$.

In $\mathbb{T}=(\mathbb{R}\cup\\{\infty\\},\min,+_{\mathbb{R}},\infty,0)$,
$a<b$ if $a$ is less than $b$ as elements of $\mathbb{R}$.

If we instead use the $(\max,+)$ semiring
$(\mathbb{R}\cup\\{-\infty\\},\max,+_{\mathbb{R}},-\infty,0)$, $a<b$ in our
semiring if $a$ is greater than $b$ as elements of $\mathbb{R}$.

Because this ordering is defined algebraically, we have the useful fact that
idempotent semiring homomorphisms are monotone with respect to this ordering.

######  Proposition 2.14.

Semiring homomorphisms between idempotent semirings are order preserving.

###### Proof.

Let $\phi:S\to S^{\prime}$ be a semiring homomorphism, and let $a\leq b$, i.e.
$a+b=a$.

$\phi(a)+\phi(b)=\phi(a+b)=\phi(a)$

i.e. $\phi(a)\leq\phi(b)$. ∎

######  Definition 2.15.

Let $X\subseteq S$ be a finite subset of an idempotent semiring $S$. We
define:

$\inf(X)=\sum_{x\in X}x$

In particular, for $a,b\in S$, we define

$\inf(a,b)=a+b$

Remark 2.16. It is equivalent to define an idempotent semiring as a unital
monoid $(S,*,1_{S})$ equipped with a lower semi-lattice order and a maximal
element $0_{S}$. The above definition gives this correspondence.

######  Definition 2.17.

If the natural partial order on an idempotent semiring $S$ is a total order
then we say that $S$ is a totally ordered idempotent semiring.

Most idempotent semirings are not totally ordered. For instance, in general
the power set of a monoid carries a non-totally ordered semiring structure:

Example 2.18.  Let $(M,*)$ be a monoid, not necessarily commutative, with
identity $e$. We can construct the power set idempotent semiring of $M$ as the
set of subsets of $M$, $2^{M}$, with addition being union and multiplication
being Minkowski multiplication. That is:

$A+B=A\cup B$ $A*B=\\{ab:a\in A,b\in B\\}$

The additive identity of $2^{M}$ is the empty set, $\emptyset$, and the
multiplicative identity is the singleton, $\\{e\\}$.

$A\leq B$ if and only if $A\supseteq B$.111When dealing with idempotent
semirings constructed from families of sets with $+=\cup$ we will often have
to deal with this reversal of order. If we chose the opposite order convention
then our partial order would appear flipped when dealing with standard
valuations. The ordering is useful shorthand and a useful property, but any
choice of order convention will make working with some semirings awkward.

######  Proposition 2.19.

$2^{M}$ is totally ordered if and only if $M=\\{e\\}$.

######  Definition 2.20.

A semiring $S$ is complete if it has infinitary addition and infinitary
distribution. i.e. for any, potentially infinite, index set $I$ there is a
summation $\sum_{I}$ such that

$\displaystyle a*\left(\sum_{i\in I}b_{i}\right)$ $\displaystyle=\sum_{i\in
I}a*b_{i}$ $\displaystyle\left(\sum_{i\in I}b_{i}\right)*a$
$\displaystyle=\sum_{i\in I}b_{i}*a.$

We require these operations to behave as one would expect with regard to
decompositions of the index sets.

$\sum_{\emptyset}b_{i}=0_{S}\qquad\sum_{\\{1\\}}b_{i}=b_{1}\qquad\sum_{\\{1,2\\}}b_{i}=b_{1}+b_{2}$

If $\bigcup I_{j}=I$ and the $I_{j}$ are pairwise disjoint then

$\sum_{j\in J}\sum_{i\in I_{j}}b_{i}=\sum_{i\in I}b_{i}.$

The above requirements can be stated concisely as: $S$ is a complete semiring
if and only if $(S,+)$ is a complete monoid.

Let be a cardinal. $S$ is complete for sums of size or less or -complete if it
has summation operations that satisfies the corresponding distributive and
decomposition laws for any index set $I$ such that $|I|\leq\beth$.

A homomorphism of complete semirings is a semiring homomorphism that respects
these summation operations, that is:

$\phi(\sum_{I}b_{i})=\sum_{I}\phi(b_{i})$

The power set semiring is a canonical example of a complete idempotent
semiring. The boolean and tropical semirings are also complete. All semirings
are complete for finite sums, and all idempotent semirings may be extended to
be complete (see [Gol02]).

With respect to the semiring ordering, a complete semiring is not necessarily
a complete lattice. We are only requiring it to have and respect potentially
infinite infinums, but it does not need to have or respect potentially
infinite supremums. In particular even if such supremums exist they may not be
preserved by complete semiring homomorphisms.

The complete subsemirings of the lattice of the power set of a set have an
important name – topologies.

Example 2.21.  Let $Z$ be a set.

Let $\tau\subseteq 2^{Z}$ be an open-topology. For any collection
$X\subseteq\tau$, $\bigcup_{x\in X}x\in\tau$ and for any finite collection
$\bigcap_{i=1}^{n}x_{i}\in\tau$, as well as $\emptyset,Z\in\tau$.

The topology $\tau$ is a subobject of the complete semiring $2^{R}$, where for
$A,B\in\tau$

$\displaystyle A+_{\tau}B$ $\displaystyle=A\cup B,$ $\displaystyle A*_{\tau}B$
$\displaystyle=A\cap B,$ $\displaystyle 0_{\tau}$ $\displaystyle=\emptyset,$
$\displaystyle 1_{\tau}$ $\displaystyle=Z.$

For a closed-topology $\tau^{\prime}$ (i.e. closed under infinite intersection
and finite unions) we flip these operations with

$\displaystyle A+_{\tau^{\prime}}B$ $\displaystyle=A\cap B,$ $\displaystyle
A*_{\tau^{\prime}}B$ $\displaystyle=A\cup B,$ $\displaystyle
0_{\tau^{\prime}}$ $\displaystyle=Z,$ $\displaystyle 1_{\tau^{\prime}}$
$\displaystyle=\emptyset.$

In both cases we have a complete semiring. By definition topologies are
exactly the complete subsemirings of $2^{Z}$, closed topologies with respect
to the ordering $A\leq B\iff A\subseteq B$ and open topologies with respect to
the dual ordering $A\leq B\iff A\supseteq B$.

Remark 2.22. Due to this fact complete, commutative, multiplicatively and
additively idempotent semirings appear often in topology under the name of
frames and coframes. A frame is a generalization of the lattice of open sets
of a topology and a coframe is a generalization of the lattice of closed sets
of a topology. Frames are the basic structure of the field of point-free
topology and a powerful generalization of topological spaces. An introduction
to point-free topology and the applications of frames is the book [Joh82].

######  Definition 2.23.

A congruence $C$ of a semiring $S$ is a subsemiring of $S\times S$ that is
also an equivalence relation.

  * •

For all $a\in S$, $(a,a)\in C$.

  * •

If $(a,b),(b,c)\in C$ then $(a,c)\in C$.

  * •

If $(a,b),(c,d)\in C$ then $(b,a),(a+c,b+d),(ac,bd)\in C$.

Congruences are closed under arbitrary intersection, which gives us the
following definition:

######  Definition 2.24.

Given a set of relations $X\subseteq S\times S$, we say that the congruence
generated by $X$, $\left\langle X\right\rangle$, is the intersection of all
congruences that contain $X$.

If our semiring $S$ is a ring then the set of congruences is in bijection with
the set of ideals, as each congruence is determined by the equivalence class
of $0$.

Congruences are exactly the equivalence relations on semirings such that the
equivalence classes form a well defined semiring under the induced operations.

######  Definition 2.25.

Given a semiring $S$ and a congruence $C$, we say the quotient $S/C$ is the
semiring on the equivalence classes of $C$ equipped with the operations
induced from $S$.

A more detailed treatment of congruences and quotient semirings can be found
in chapter 8 of [Gol99].

It is important to note that for general semirings the quotients of a semiring
$S$ are not determined by the equivalence class of $0$. There are often many
nontrivial quotients of a semiring such that the equivalence class of $0$ is
just $\\{0\\}$.

######  Definition 2.26.

Given a semiring $S$, and a set of variables $X$, we say that the semiring of
expressions over $X$, denoted $S\left\langle X\right\rangle$, and also called
the non-commutative polynomial semiring, is the freely generated semiring over
elements in $S$ and variables in $X$ with no relations between them apart from
defining $1_{S}=1_{S\left\langle X\right\rangle}$ and $0_{S}=0_{S\left\langle
X\right\rangle}$.

So for instance we have elements in $\mathbb{T}\left\langle x,y\right\rangle$
of the form:

$3xy+12x+x12+0$

where $12x\neq x12$ and $xy\neq yx$.

######  Definition 2.27.

The polynomial semiring over variables $X$ is the quotient

$S\left\langle X\right\rangle/\sim$

where $\sim$ is the congruence generated by relations specifying that
variables from $X$ commute with one another and with elements in $S$. These
relations are

  * •

$xy\sim yx$ for $x,y\in X$,

  * •

$sx\sim xs$ for $s\in S,x\in X$.

We denote this semiring $S[X]$.

$S\left\langle X\right\rangle$ is the semiring of algebraic expressions where
we do not assume that our variables commute with $S$ or with each other,
whereas $S[X]$ is the semiring where we assume that our variables do commute
with $S$ and with each other.

######  Proposition 2.28.

Let $\Gamma$ be an idempotent semiring, $X$ any set, and $C$ any congruence on
$\Gamma$. The following are all idempotent semirings.

  * •

$\Gamma\left\langle X\right\rangle$

  * •

$\Gamma[X]$

  * •

$C$

  * •

$\Gamma/C$

This follows from Proposition 2.4, as $1_{\Gamma}+1_{\Gamma}=1_{S}$ for each
of these semirings.

A detailed reference on the general theory of semirings is the book [Gol99].

###  2.2 Valuations

######  Definition 2.29.

Let $R$ be a ring and $\Gamma$ an idempotent semiring. We say that a function
$\nu:R\to\Gamma$ is a valuation if $\nu$ is:

Unital:

$\nu(0_{R})=0_{\Gamma}$, $\nu(1_{R})=1_{\Gamma}$,

Multiplicative:

$\nu(a*_{R}b)=\nu(a)*_{\Gamma}\nu(b)$,

Superadditive:

$\nu(a+_{R}b)\geq\nu(a)+_{\Gamma}\nu(b)=\inf_{\Gamma}(\nu(a),\nu(b))$.

Invariant under negation:

$\nu(-1_{R})=\nu(1_{R})$

From Definition 2.12 we get that superadditivity is equivalent to the
following identity:

$\nu(a+_{R}b)+_{\Gamma}\nu(a)+_{\Gamma}\nu(b)=\nu(a)+_{\Gamma}\nu(b).$

The above definition aligns with the generalized semiring valuation in [GG16].

Remark 2.30. You can recover the traditional definition of a valuation in
arithmetic number theory by taking $\Gamma=\mathbb{T}$. A Krull valuation is a
valuation where that the canonical order on $\Gamma$ is a total order and
$\Gamma$ is commutative.

Example 2.31.  Let
$\Gamma=(\mathbb{Z}\cup\\{\infty\\},\min,+_{\mathbb{Z}},\infty,0)$.

Fix a prime number $p$. Each $x\in\mathbb{Q}$ can be written in the form
$p^{n}*\frac{a}{b}$ for integers $a,b$ and $n$, where $p$ does not divide $a$
or $b$. The $p$-adic valuation on $\mathbb{Q}$ is the map:

$\nu_{p}(x)=n$

$\nu_{p}:\mathbb{Q}\to\Gamma$ forms a valuation.

The following lemma also appears in [GG16] as Lemma 2.5.3:

######  Lemma 2.32.

Let $R$ be a ring, and let $\nu:R\to\Gamma$ be a valuation on $R$.

For any $a,b\in R$ the following three values are equal:

  * •

$\nu(a)+\nu(b)$

  * •

$\nu(a+b)+\nu(a)$

  * •

$\nu(a+b)+\nu(b)$

Remark 2.33. The above theorem is not just a consequence of superadditivity,
but is in fact equivalent to superadditivity.

This theorem is the generalized version of a common theorem on arithmetic and
Krull valuations. If $\nu$ is a Krull valuation and $\nu(a)\neq\nu(b)$ then
$\nu(a+b)=\min(\nu(a),\nu(b))$. In the case where $\Gamma$ is totally ordered
this is a corollary of the above theorem, however for a non-totally ordered
codomain the above statement does not imply that
$\nu(a+b)=\min(\nu(a),\nu(n))$ instead we just have that the infinum of any
two elements of $\\{\nu(a),\nu(b),\nu(a+b)\\}$ are the same.

######  Corollary 2.34.

Let $a$ and $b$ be rational numbers with $a\geq b$.

$\operatorname{gcd}(a,b)=\operatorname{gcd}(a-b,b)$

###### Proof.

To prove this we just note that we have a valuation from the ring $\mathbb{Q}$
into the idempotent semiring $(\mathbb{Q}_{\geq
0},\operatorname{gcd},*,0,1)$:222Here we define that for
$\frac{a}{b},\frac{c}{d}\in\mathbb{Q}_{\geq 0}$:
$\operatorname{gcd}(\frac{a}{b},\frac{c}{d})=\frac{\operatorname{gcd}(ad,bc)}{bd}$
and $\operatorname{gcd}(0,x)=x$ for all $x\in\mathbb{Q}_{\geq 0}$. One can
check that the above definition does not depend on a choice of representation
for our fractions.

$x\mapsto|x|$

Lemma 2.32 tells us:

$\operatorname{gcd}(|a-b|,|-b|)=\operatorname{gcd}(|a|,|-b|).$

∎

Remark 2.35. We note that due to our invariance under negation and Lemma 2.32,
we can also state:

$\nu(a-b)+\nu(b)=\nu(a)+\nu(-b)=\nu(a)+\nu(b).$

##  3 Universal Valuations

For a fixed ring $R$ we can examine the slice category
$R\downarrow\mathbf{ISR}$ where objects are valuations $\nu:R\to\Gamma$ and
arrows are given by semiring homomorphisms that make the following triangle
commute.

${R}$${\Gamma}$${{\Gamma^{\prime}}}$$\scriptstyle{\nu}$$\scriptstyle{\nu^{\prime}}$$\scriptstyle{\varphi}$

Proposition 2.5.4 from [GG16] can be stated as the following.

######  Proposition 3.1.

$R\downarrow\mathbf{ISR}$ has an initial object, $R\to\Gamma_{R}$.

As $R\to\Gamma_{R}$ is initial in $R\downarrow\mathbf{ISR}$, it factors all
other valuations of $R$. For any valuation $\nu:R\to\Gamma$ there exists a
unique semiring homomorphism $\Gamma_{R}\to\Gamma$ such that the composite
$R\to\Gamma_{R}\to\Gamma$ is $\nu$.

That is, $\mathbf{ISR}(\Gamma_{R},\Gamma^{\prime})$ is in bijection with the
collection of generalized valuations on $R$ with codomain $\Gamma^{\prime}$.

Remark 3.2. For the rest of this paper we will often consider valuations over
complete idempotent semirings, and consider universal valuation semirings that
are universal over valuations whose codomain are complete. Here we can take
advantage that we can freely complete semirings, [Gol02], and that this
completion functor is a left adjoint and so it preserves colimits. In
particular this means that the universal complete valuation semiring is the
completion of the standard universal valuation semiring.

[GG16] gives a construction of $\Gamma_{R}$, the definition we give below is
the one in [Ber23] that is a very slight modification to apply to potentially
non-commutative rings.333The form in [GG16] is the same quotient, but the free
semiring is the polynomial semiring $\mathbb{B}[x_{R}]$ rather than
$\mathbb{B}\left\langle x_{R}\right\rangle$. $\mathbb{B}\left\langle
x_{R}\right\rangle$ is the free idempotent semiring on the set $R$ whereas
$\mathbb{B}[x_{R}]$ is the free commutative idempotent semiring on the set
$R$. If $R$ is commutative the quotient forces $\Gamma_{R}$ to be as well, so
when considering commutative rings one may restrict to the category of
commutative idempotent semirings.

######  Definition 3.3.

We construct the universal valuation semiring of $R$, $\Gamma_{R}$ as follows.

For each $a\in R$, we define a variable $x_{a}$. We denote the free semiring
of boolean expressions over these variables as

$\mathbb{B}\left\langle x_{R}\right\rangle$

We then define $\Gamma_{R}$ as

$\Gamma_{R}=\mathbb{B}\left\langle x_{R}\right\rangle/\sim$

where $\sim$ is the congruence generated by the following relations.

$\displaystyle x_{0}$ $\displaystyle\sim 0$ $\displaystyle x_{1}$
$\displaystyle\sim 1$ $\displaystyle x_{-1}$ $\displaystyle\sim 1$
$\displaystyle x_{a}x_{b}$ $\displaystyle\sim x_{ab}$ $\displaystyle
x_{a+b}+x_{a}+x_{b}$ $\displaystyle\sim x_{a}+x_{b}$

The valuation associated with this semiring, is the map

$\nu(a)=[x_{a}].$

Example 3.4.  This is example 8 from [Ber23].

Let $R$ be the ring of upper triangular $2\times 2$ matrices over
$\mathbb{F}_{2}$. $R$ has eight elements and they are generated by the
matrices:

$i=\begin{bmatrix}1&0\\\ 0&0\end{bmatrix}\qquad j=\begin{bmatrix}0&1\\\
0&0\end{bmatrix}\qquad k=\begin{bmatrix}0&0\\\ 0&1\end{bmatrix}$

$\Gamma_{R}$ consists of $\mathbb{B}$ linear combinations of the elements:
$0,1,x_{i},x_{j},x_{k},x_{i+j},x_{j+k},x_{i+j+k}$. Note that
$x_{i+k}=x_{1}=1$.

The multiplicative structure of $\Gamma_{R}$ is generated by the following
multiplication table.

| $x_{i}$ | $x_{j}$ | $x_{k}$ | $x_{i+j}$ | $x_{j+k}$ | $x_{i+j+k}$
---|---|---|---|---|---|---
$x_{i}$ | $x_{i}$ | $x_{j}$ | $0$ | $x_{i+j}$ | $x_{j}$ | $x_{i+j}$
$x_{j}$ | $0$ | $0$ | $x_{j}$ | $0$ | $x_{j}$ | $x_{j}$
$x_{k}$ | $0$ | $0$ | $x_{k}$ | $0$ | $x_{k}$ | $x_{k}$
$x_{i+j}$ | $x_{i}$ | $x_{j}$ | $x_{j}$ | $x_{i+j}$ | $0$ | $x_{i}$
$x_{j+k}$ | $0$ | $0$ | $x_{j+k}$ | $0$ | $x_{j+k}$ | $x_{j+k}$
$x_{i+j+k}$ | $x_{i}$ | $x_{j}$ | $x_{j+k}$ | $x_{i+j}$ | $x_{k}$ | $1$

The additive structure has the following diagramatic form.

${1}$${{x_{k}}}$${{x_{i}}}$${{x_{i+j+k}}}$${{x_{j+k}}}$${{x_{i+j}}}$${{x_{j}}}$

The sum of any two elements with the same lines type and color between them
are the same item. So, for instance, we have

$[x_{i}+x_{k}]=[x_{k}+1]=[1+x_{i}]$

and

$[1+x_{i+j+k}]=[x_{i+j+k}+x_{j}]=[1+x_{j}].$

There is a single element corresponding to a non-degenerate sum of three
elements, and it is obtained by the sum of any three elements such that the
lines between them are distinct colors or types, e.g.:

$[x_{i}+x_{j}+x_{k}]=[1+x_{i+j}+x_{j}]$

But, for instance, the element $[1+x_{i+j}+x_{j+k}]$ is equal to
$[1+x_{i+j}]$, but is not equal to $[x_{i}+x_{j}+x_{k}]$.

Theorem 8 in [Ber23] classifies the additive structure of these universal
valuation semirings.

######  Theorem 3.5 (Structure Theorem for $\Gamma_{R}$).

Let $(a_{i})_{i\in I}$ and $(b_{j})_{j\in J}$ be finite collections of
elements in a ring $R$. In $\Gamma_{R}$ we have:

$\left[\sum_{i\in I}x_{a_{i}}\right]=\left[\sum_{j\in J}x_{b_{j}}\right]$

if and only if
${\operatorname{Span}}_{\mathbb{Z}}(a_{i})={\operatorname{Span}}_{\mathbb{Z}}(b_{j})$.

Remark 3.6. This theorem can be phrased another way.

Consider $R$ as a $\mathbb{Z}$-module, and let
$\operatorname{SubMod}_{\mathbb{Z}}(R)$ be the collection of $\mathbb{Z}$
submodules of $R$. We note that if $A_{i}$ is an arbitrary collection of
$\mathbb{Z}$-submodules of $R$ then $\bigcap
A_{i}\in\operatorname{SubMod}_{\mathbb{Z}}(R)$. From this we can define that
for $X\subseteq R$ the submodule generated by $X$ is:

$\left\langle X\right\rangle=\bigcap_{X\subseteq
A\in\operatorname{SubMod}_{\mathbb{Z}}(R)}A$

We then note that $\operatorname{SubMod}_{\mathbb{Z}}(R)$ has an idempotent
semiring structure, with:

$\displaystyle A+_{\operatorname{SubMod}_{\mathbb{Z}}(R)}B$
$\displaystyle=\left\langle A\cup B\right\rangle$ $\displaystyle
A*_{\operatorname{SubMod}_{\mathbb{Z}}(R)}B$ $\displaystyle=\left\langle
ab:a\in A,b\in B\right\rangle$ $\displaystyle
0_{\operatorname{SubMod}_{\mathbb{Z}}(R)}$ $\displaystyle=\left\langle
0_{R}\right\rangle=\\{0_{R}\\}$ $\displaystyle
1_{\operatorname{SubMod}_{\mathbb{Z}}(R)}$ $\displaystyle=\left\langle
1_{R}\right\rangle$

Let $\operatorname{SubMod}^{0}_{\mathbb{Z}}(R)$ be the collection of finitely
generated $\mathbb{Z}$ submodules of $R$. We note that this is a subsemiring
of $\operatorname{SubMod}_{\mathbb{Z}}(R)$. There is a surjective valuation
$R\to\operatorname{SubMod}^{0}_{\mathbb{Z}}(R)$ sending $r$ to the subspace
generated by $r$, $\left\langle r\right\rangle$. Theorem 3.5 states that the
induced factoring map between $\Gamma_{R}$ and
$\operatorname{SubMod}^{0}_{\mathbb{Z}}(R)$ is injective as well as
surjective, and so $\Gamma_{R}\cong\operatorname{SubMod}^{0}_{\mathbb{Z}}(R)$.
We further note that if we take our universal valuation to be the limit over
valuations whose codomain are complete idempotent semirings, then our
universal object would be $\operatorname{SubMod}_{\mathbb{Z}}(R)$.

In the remainder of this paper we will classify the universal object
associated with two more classes of valuations: positive valuations and
positive multiplicatively idempotent valuations. In the same way that the
universal valuation semiring defines $\operatorname{SubMod}_{\mathbb{Z}}(R)$
up to isomorphism, we will see that these two additional classes of valuations
define the semirings of ideals of a ring, as well as the closed Zariski
topology on the spectrum of a ring up to isomorphism.

##  4 Positive Valuations

Conventions: While all of the results in [Ber23] hold for commutative and non-
commutative rings, for the rest of this paper we will restrict to commutative
rings.

We will be particularly interested in the product of principally generated
ideals, and for noncommutative rings it is not in general true that
$(a)*(b)=(ab)$. As a result of this the map $r\mapsto(r)$ is not guaranteed to
be multiplicative in the noncommutative case.

######  Definition 4.1.

A valuation $\nu:R\to\Gamma$ is positive if for each $r\in R$, $\nu(r)\geq
1_{\Gamma}$

Remark 4.2. When $\Gamma=\mathbb{T}$, this is saying that for all $r\in R$,
$\nu(r)\geq 0\in\mathbb{R}$.

######  Proposition 4.3.

Let $\nu:R\to\Gamma$ be any valuation. The set $\\{r\in R:\nu(r)\geq
1_{\Gamma}\\}$ is a subring of $R$.

######  Definition 4.4.

For a given valuation $\nu:R\to\Gamma$ we call the subring $\\{r\in
R:\nu(r)\geq 1_{\Gamma}\\}$ the $\nu$-positive subring of $R$. When $\nu$ is
understood we denote this $R_{+}$.

Remark 4.5. $R_{+}$ is the largest subring of $R$ where $\nu$ forms a positive
valuation.

######  Proposition 4.6.

The $\nu$-positive subring of the universal valuation semiring $\Gamma_{R}$ is
exactly the image of the canonical map $\mathbb{Z}\to R$, sending
$1_{\mathbb{Z}}$ to $1_{R}$.

###### Proof.

This is a consequence of our structure theorem, theorem 3.5, as if $x_{r}\geq
1=x_{1}$ then $r\in{\operatorname{Span}}_{\mathbb{Z}}(1_{R})$. ∎

It is well known classically that in certain contexts the existence of
positive valuations are equivalent to deep statements on the structure of the
ideals of a commutative ring.

######  Definition 4.7.

Let $R$ be a integral domain and let $K=\operatorname{Frac}(R)$.

If there exists a valuation $\nu:K\to\Gamma$ where $\Gamma$ is a totally
ordered idempotent semiring and $K_{+}=R$, then we say that $R$ is a valuation
ring.

A classical result states:

######  Theorem 4.8.

If a domain is a valuation ring then its ideals are totally ordered by
inclusion.

###### Proof.

We reproduce the classical approach.

Given a totally ordered valuation $\nu:K\to\Gamma$ then for all $x,y\in K$
either $xy^{-1}\geq 1$ giving us $xy^{-1}\in K_{+}=R$ or $xy^{-1}\leq 1$
giving us $yx^{-1}\in R$. If $xy^{-1}\in R$ then $(y)\subseteq(x)$ and if
$yx^{-1}\in R$ then $(x)\subseteq(y)$. If the principal ideals are totally
ordered then one can show that arbitrary ideals are totally ordered. ∎

This converse of this statement is also true and is a theorem by Krull,
[Kru39].

Remark 4.9. For the above proof to work we just needed that for all $x,y\in K$
either $\nu(xy^{-1})\leq 1_{\Gamma}$ or $\nu(xy^{-1})\geq 1_{\Gamma}$. The
fact that $\Gamma$ is totally ordered gives us this, but we can generalize
this considerably.

######  Proposition 4.10.

Let $R$ be an integral domain and $K=\operatorname{Frac}(R)$.

If there exists a valuation $\nu:K\to\Gamma$ such that for all $k\in K$,
$\nu(k)$ is comparable to $1_{\Gamma}$, and $K_{+}=R$, then the ideals of $R$
are totally ordered by inclusion.

While this result seems to be algebraic trickery, it suggests that there is a
relationship between the space of positive valuations and the structure of the
ideals of a semiring. This connection turns out to be a deep, and arguably
defining, connection. We will formalize this connection, showing that every
positive valuation (with complete codomain) admits a Galois connection with
the lattice of ideals of a ring $R$, as well as showing that the semiring of
ideals of a ring $R$ can be identified as the universal positive valuation
that factors all other positive valuations with complete codomain.

Just as there is a universal valuation semiring, there is a universal positive
valuation semiring that factors all positive valuations.

######  Definition 4.11.

Given a ring $R$, the universal positive valuation semiring, $\Gamma_{R}^{+}$
is the semiring:

$\Gamma_{R}/\left\langle x_{r}+1=1\right\rangle.$

######  Proposition 4.12.

This semiring, equipped with the valuation $\nu(r)=[r]$, is initial among the
positive valuations of $R$.

######  Definition 4.13.

Given a (commutative or non-commutative) ring $R$, the set of two sided ideals
of $R$, $I(R)$ has an idempotent semiring structure, with

$I+J=\\{a+b:a\in I,b\in J\\},$ $I*J=\left(\sum_{i=0}^{N}a_{i}b_{i}:a_{i}\in
I,b_{i}\in J,N\in\mathbb{N}\right).$

In general $\sum I_{j}$ is the ideal generated by finite sums of elements in
the indexed ideals, and $\prod_{j=1}^{N}I_{j}$ is the ideal generated by
finite sums of elements of the form $a_{1}*a_{2}*...*a_{N}$ for $a_{j}\in
I_{j}$.

Under this structure we have:

$1_{I(R)}=(1),\quad 0_{I(R)}=(0)$ $I\leq J\iff I\supseteq J.$

We call this the semiring of ideals of $R$.

######  Definition 4.14.

When $R$ is commutative we may form a positive valuation $\iota:R\to I(R)$
under the mapping

$\iota:x\mapsto(x).$

######  Theorem 4.15.

Let $\nu:R\to\Gamma$ be a positive valuation, and let $\Gamma$ be a complete
semiring444Or at least closed under sums of cardinality $|R|$ or less.. Then
there exists a monotone Galois connection between $I(R)$ and $\Gamma$.

That is, there exist monotone maps $N:I(R)\to\Gamma$ and $I:\Gamma\to I(R)$
such that

$\gamma\leq N(J)\iff I(\gamma)\leq J$

for $\gamma\in\Gamma$, $J\in I(R)$.

###### Proof.

For each $\gamma\in\Gamma$ we first check that the set $I(\gamma)=\\{r\in
R:\nu(r)\geq\gamma\\}$ forms an ideal.

  * •

For all $\gamma$, $0_{R}\in I(\gamma)$.

  * •

If $a,b\in I(\gamma)$ then $\nu(a-b)\geq\nu(a)+\nu(b)\geq\gamma$, and so
$a-b\in I(\gamma)$.

  * •

Let $a\in I(\gamma)$ and $r\in R$. $\nu(ra)=\nu(r)\nu(a)\geq
1_{\Gamma}*\gamma=\gamma$ and so $ra$ and similarly $ar$ are in $I(\gamma)$.

The map $\gamma\mapsto I(\gamma)$ is not quite a semiring homomorphism as it
is not necessarily additive, but it is a monic map and if $R$ is a PID or a
commutative Noetherian ring then it is guaranteed to be multiplicative.

Similarly for any ideal $I$, as $\Gamma$ is complete we can define the map $N$
by $N(I)=\sum_{x\in I}\nu(x)$.

We finally prove that $\gamma\leq N(J)$ if and only if $I(\gamma)\leq J$.

Let $\gamma\leq N(J)$, then for all $x\in J$ we have that $\nu(x)\geq\gamma$
and so $x\in I(\gamma)$. Therefore $I(\gamma)\supseteq J$ or $I(\gamma)\leq J$
by Definition 2.12.

Let $I(\gamma)\leq J$, or similarly $J\subseteq I(\gamma)$, then for all $x\in
J$ we have that $\nu(x)\geq\gamma$, and so $\sum_{x\in J}\nu(x)\geq\sum_{x\in
J}\gamma=\gamma$. ∎

Remark 4.16. This Galois connection means we can form an idempotent closure
operator on $\Gamma$ by

$\gamma\mapsto N(I(\gamma)).$

Note that $\gamma\leq N(I(\gamma))$ always holds.

We can also form an idempotent kernel operator on $I(R)$ by

$J\mapsto I(N(J))$

We similarly also have that $I(N(J))\leq J$ or $I(N(J))\supseteq J$.

Example 4.17.  Consider $\mathbb{Z}$ equipped with the $p$-adic valuation.

$\mathbb{Z}$ is a PID, and we get that for $(x)\subseteq Z$, $N((x))=\nu(x)$.

Our kernel operator is $I(N(x))=(p^{\nu(x)})$, our closure operator
$N(I(a))=N((p^{a}))=a$ for all $a$. Under the $3$-adic valuation we get that
$I(N((2))=\mathbb{Z}$, but $I(N((6)))=(3)=I(N((24)))$.

######  Theorem 4.18.

For any positive valuation $\nu:R\to\Gamma$ for which $\Gamma$ is complete,
the induced map $N:I(R)\to\Gamma$ is a complete semiring homomorphism whuch
factors $\nu$. That is $\nu=N\circ\iota$, where $\iota$ is the valuation
$\iota(r)=(r)$ on $I(R)$.

###### Proof.

While the map $I$ does not always form a semiring homomorphism, we can see
that $N$ does, as

$N(I+J)=\sum_{x+y,x\in I,j\in J}\nu(x+y)\geq\sum_{x+y,x\in I,j\in
J}\nu(x)+\nu(y)=\sum_{x\in I}\nu(x)+\sum_{y\in J}\nu(y).$

As $0\in I\cap J$, and we have that in an idempotent semiring sums are always
less than or equal to subsums, we have

$\sum_{x+y,x\in I,j\in J}\nu(x+y)\leq\sum_{x\in I}\nu(x)$

and

$\sum_{x+y,x\in I,j\in J}\nu(x+y)\leq\sum_{y\in J}\nu(y)$

and so due to additive idempotence we get

$\sum_{x+y,x\in I,j\in J}\nu(x+y)\leq\sum_{x\in I}\nu(x)+\sum_{y\in J}\nu(y)$

and so $N$ forms an additive homomorphism. By finite superadditivity and the
fact that for an infinite sum, elements in $\sum J_{i}$ are finite sums of
elements in $J_{i}$’s, we can see that $N$ also respects the infinitary sums
of $I(R)$.

We have that by superadditivity

$N(I*J)\geq\sum_{a\in I,b\in J}\nu(a)\nu(b)=\sum_{a\in I}\nu(a)*\sum_{b\in
J}\nu(b),$

however as $I*J\supseteq\\{ab:a\in I,b\in J\\}$ we have that $\sum_{a\in
I}\nu(a)*\sum_{b\in J}\nu(b)$ is a subsum of $N(I*J)$ and thus is greater than
or equal to it. Thus $N(I*J)=N(I)*N(J)$.

$N$ forms a map of positive valuations. Let $\nu:R\to\Gamma$ be a positive
valuation. As $\nu(r)\geq 1_{\Gamma}$ for all $r\in\Gamma$ then for any
$x\in\Gamma$ we have $\nu(rx)=\nu(r)\nu(x)\geq\nu(x)$. Thus for any any
principal ideal we have $N((x))=\nu(x)$, and so $\nu=N\circ\iota$. ∎

Remark 4.19. Once we know that the map $N:I(R)\to\Gamma$ exists it is not
suprising that there exists a Galois correspondence. Complete idempotent
semiring homomorphisms are meet preserving maps from posets that have all
meets, and so by the adjoint functor theorem for posets every complete
idempotent semiring homomorphism admits an adjoint. The nLab article [nLa24]
has more details on this construction.

It is suprising however that the specific adjoint of $N:I(R)\to\Gamma$ is
cleanly given by the preimage of the upward closure of an element in $\Gamma$:
$I(\gamma)=\nu^{-1}\left(\\{\rho:\rho\geq\gamma\\}\right)$

Remark 4.20. We note that if $\\{a_{i}\\}\subseteq R$ generate the ideal $J$
then $N(J)=\sum a_{i}$.

If $x\in J$ we can write it as a finite sum $x=\sum r_{i}a_{i}$. By
superadditivity and positivity of $\nu$ we get

$\nu(x)\geq\sum\nu(r_{i})\nu(a_{i})\geq\sum\nu(a_{i}).$

The behavior of $N$ is entirely determined by the valuation on generating sets
of our ideals.

######  Corollary 4.21.

$I(R)$ is initial among positive valuations whose valuation semiring is
complete for sums of size $|R|$ or less.555Strictly speaking: Let  be the
largest cardinality of a minimal generating set of an an ideal of $R$, we can
define the map $N:I(R)\to\Gamma$ as long as $\Gamma$ is closed under sums of
cardinality  or less.

If $R$ is Noetherian then $I(R)$ is initial among all positive valuations.

Here we note that if $R$ is Noetherian then every ideal is finitely generated,
and so we can reduce every sum, $N(I)$, to a finite sum.

######  Corollary 4.22.

If $R$ is Noetherian, then $I(R)\cong\Gamma_{R}^{+}=\Gamma_{R}/\left\langle
x_{r}+1=1\right\rangle$

Example 4.23.  We will briefly show using example 3.3 why our assumption of
commutativity is necessary. Here $R$ is the subring of $M_{2}(\mathbb{Z}_{2})$
consisting of upper triangular matrices. We will name elements of $R$ as:

$i=\begin{bmatrix}1&0\\\ 0&0\end{bmatrix}\qquad j=\begin{bmatrix}0&1\\\
0&0\end{bmatrix}\qquad k=\begin{bmatrix}0&0\\\ 0&1\end{bmatrix}$

Here we note that $R$ is finite and every semiring is closed under finite
sums, and so if our theorem held without requiring commutativity we would have
$\Gamma_{R}^{+}\cong I(R)$.

The elements of $I(R)$ are the two sided ideals:

$\displaystyle(0)$ $\displaystyle=\\{0\\}$ $\displaystyle(i)=(i+j)$
$\displaystyle=\\{0,i,j,i+j\\}$ $\displaystyle(j)$ $\displaystyle=\\{0,j\\}$
$\displaystyle(k)=(j+k)$ $\displaystyle=\\{0,j,k,j+k\\}$
$\displaystyle(1)=(i+k)=(i+j+k)$ $\displaystyle=R$

The multiplication table of $I(R)$ is the following.

| $(i)$ | $(j)$ | $(k)$
---|---|---|---
$(i)$ | $(i)$ | $(j)$ | $(j)$
$(j)$ | $(0)$ | $(0)$ | $(j)$
$(k)$ | $(0)$ | $(0)$ | $(k)$

And the additive structure is given by the following lattice, with an arrow
$a\to b$ meaning $a\leq b$.

${{(0)}}$${{(j)}}$${{(i)}}$${{(k)}}$${{(1)}}$

We note that this is indeed the structure one would expect when quotienting
just the additive structure of $\Gamma_{R}$ from Example 3.3 by the congruence
generated by $x_{r}+1=1$ for all $r\in R$. However this does not form a
valuation as the map $r\mapsto(r)$ is not multiplicative: $i*_{R}k=0$ but
$(i)*_{I(R)}(k)=(j)$. The initial positive valuation of $R$ in this case is
not $I(R)$ but $I(R)/\left\langle(j)\sim 0\right\rangle$.

A consequence of Corollary 4.21 is that the complete idempotent semiring of
ideals of a ring can be characterized up to isomorphism via the category of
positive $R$-valuations whose codomain is complete. Many properties of the
ideals of $R$ can be extracted from the structure of this semiring, even
without reference to the underlying sets. For instance one can derive the
collection of prime ideals of a ring purely from the semiring structure.

######  Definition 4.24.

Let $\Gamma$ be an idempotent semiring.

We say $p\in\Gamma$ is a prime element if $ab\geq p$ implies $a\geq p$ or
$b\geq p$.

######  Proposition 4.25.

$J\in I(R)$ is a prime element if and only if $J$ is a prime ideal.

Proposition 4.25 is exactly Krull’s definition of a prime ideal for a possibly
noncommutative ring.

######  Definition 4.26.

Let $\Gamma$ be an idempotent semiring. We say $F\subseteq\Gamma$ is a filter
if it satisfies the following

  * •

$F$ is nonempty.

  * •

$F$ is upward closed: for each $x\in F$, $y\in\Gamma$ $x\leq y$ implies that
$y\in F$.

  * •

$F$ is downward directed: for every $x,y\in F$, there is some $z\in F$ such
that $x\geq z$ and $y\geq z$.

This third point in particular implies that for any $x,y\in F$ we have $x+y\in
F$.

A filter is prime if $xy\in F$ implies $x\in F$ or $y\in F$.

A filter is radical if it can be written as an intersection of prime filters.

A filter is principal if it is of the form $\overline{x}=\\{y\in\Gamma:y\geq
x\\}$ for some $x\in\Gamma$.

An filter is maximal if it is proper and there is no proper filter containing
it. Maximal filters are also called ultrafilters.

These definitions, with the exception of primality, are the standard order
theoretic definitions. When our idempotent semiring is a distributive lattice
– meaning $ab=a\vee b$ – our definition of a prime element and a prime filter
are exactly the standard ones.

######  Proposition 4.27.

A principal filter $\overline{x}$ is prime if and only if $x$ is a prime
element.

######  Proposition 4.28.

$J\in I(R)$ is a radical ideal if and only if $\overline{J}$ is a radical
filter.

We note here that with Proposition 4.27, the fact that the principal filter of
radical ideal is radical may seem trivial, the converse is not necessarily
simple. It is important to note that a prime filter is not necessarily a prime
principal filter, and so the statement that $\overline{J}$ is a radical filter
does not directly state that $J$ is the intersection of prime ideals.

###### Proof.

Let $J=\sqrt{J}$ then we may write $J=\bigcap P_{i}$. We note that if $K\in
I(R)$ is such that $K\subseteq J$ (i.e. $K\geq J$) then $K\subseteq P_{i}$ for
all $i\in I$, and so $K$ is contained in the principal prime filter
$\overline{P_{i}}$ for all $i$. Similarly if $K\in\overline{P_{i}}$ for all
$i$ then $K\subseteq P_{i}$ for all $i$ and so $K\subseteq\bigcap P_{i}=J$,
Thus we get $\overline{J}=\bigcap_{i\in I}\overline{P_{i}}$.

Let $\overline{J}=\bigcap_{i\in I}F_{i}$ for $F_{i}$ prime filters. Then we
note that $J\in F_{i}$ for all $i$, and for all $x\in J$ we have $(x)\in
F_{i}$ for all $i$. Let $a^{n}\in J$, then $(a^{n})=(a^{n-1})*(a)\in F_{i}$
for all $i$. As these are prime we get that $(a)\in F_{i}$ for all $i$, and so
$(a)\in\overline{J}$. Thus we have $(a)\geq J$ or $(a)\subseteq J$, thus $a\in
J$ and we know that $J$ is radical. ∎

Primality of elements and the above properties of filters are invariant under
semiring isomorphisms, which means that we can reconstruct the prime spectrum
and the Zariski topology of a commutative ring purely from the semiring
resulting from the universal property given in Corollary 4.21.

##  5 Positive, Multiplicatively Idempotent Valuations

What if we wanted to assign a set or some lattice valued structure to each
element of a ring?

Example 5.1.  Let $R=k[x_{1},...,x_{n}]$ for $k$ an algebraically closed
field.

Consider the map:

$f\mapsto\\{\vec{x}:f(\vec{x})=0_{k}\\}$

This forms a positive, multiplicatively idempotent valuation into the semiring
$\mathcal{P}(k^{n})$, where for $X,Y\subseteq k^{n}$ we have:

$\displaystyle X+Y$ $\displaystyle=X\cap Y,$ $\displaystyle X*Y$
$\displaystyle=X\cup Y,$ $\displaystyle X\leq Y$ $\displaystyle\iff X\subseteq
Y,$ $\displaystyle 0$ $\displaystyle=k^{n},$ $\displaystyle 1$
$\displaystyle=\emptyset.$

The induced Galois correspondence $I(R)\vdash\mathcal{P}(k^{n})$ is given by
the maps below.

$\displaystyle N:I(R)$ $\displaystyle\to\mathcal{P}(k^{n})$ $\displaystyle
N:J$ $\displaystyle\mapsto\bigcap_{j\in J}\\{\vec{x}:j(\vec{x})=0\\}$
$\displaystyle I:\mathcal{P}(k^{n})$ $\displaystyle\to I(R)$ $\displaystyle
I:X$ $\displaystyle\mapsto\\{f:f(\vec{x})=0\,\forall x\in X\\}$ $\displaystyle
IN(J)$ $\displaystyle=\sqrt{J}$

This is exactly the standard Galois correspondence in algebraic geometry
between ideals and varieties. The statement that $IN(J)=\sqrt{J}$ is famously
known as Hilbert’s Nullstellensatz.

We might ask ourselves if Hilbert’s Nullstellensatz is a sort of universal
result, or arises out of some universal result on positive multiplicatively
idempotent valuations. We will show that this radical closure is the finest
closure that can arise out of a positive multiplicatively idempotent
valuation.

######  Definition 5.2.

Let $\nu:R\to\Gamma$ be a positive valuation. We say that $\nu$ is
multiplicatively idempotent if $\Gamma$ is, that is, if for all
$\gamma\in\Gamma$ we have

$\gamma*_{\Gamma}\gamma=\gamma.$

Remark 5.3. A positive multiplicatively idempotent valuation is equivalent to
a notion de zéros introduced by André Joyal in [Joy76]. In this note he states
that the prime spectrum of a ring factors all notions of zeroes. We will make
this statement precise. An explanation of the consequences of this statement
can be found in a Mathematics Stack Exchange answer by Zhen Lin Low, [Low].

To make this statement precise we need to restrict to the case where $I(R)$ is
initial, see Corollary 4.21. For this to hold we either consider slices over
the category of $|R|$-complete idempotent semirings or we assume that $R$ is
Noetherian.

Remark 5.4. In the field of point-free topology, a complete, commutative,
multiplicatively and additively idempotent semiring has a punchier name, it is
a coframe. Coframes can be thought of as a generalization of closed topologies
– every closed topology is a coframe.

Our positive, multiplicatively idempotent valuations can be seen as finding
maps $\nu:R\to\Gamma$ from a commutative ring $R$ to a coframe or closed
topology $\Gamma$ such that

$\displaystyle\nu(0)$ $\displaystyle=\top,$ $\displaystyle\nu(1)$
$\displaystyle=\bot,$ $\displaystyle\nu(a+b)$
$\displaystyle\supseteq\nu(a)\wedge\nu(b),$ $\displaystyle\nu(ab)$
$\displaystyle=\nu(a)\vee\nu(b).$

If $\Gamma$ is a topology $\top$ is the entire set and $\bot=\emptyset$.

######  Proposition 5.5.

Let $R\overset{\nu}{\to}\Gamma$ be a positive, multiplicatively idempotent
valuation. For any ideal $J\subseteq R$ we get:

$IN(J)\supseteq\sqrt{J}$

###### Proof.

As $\Gamma$ is multiplicatively idempotent we have that $\nu(f)=\nu(f^{n})$
for all $n$. Then $\nu(f)\geq\gamma$ if and only if for any $n$ we have
$\nu(f^{n})\geq\gamma$.

If $r^{n}\in I(\gamma)$, then $\nu(r^{n})=\nu(r)\geq\gamma$ and so $r\in
I(\gamma)$. ∎

For any commutative ring any positive multiplicatively idempotent valuation
gives a closure operator on the ideals of that ring that is “lower bounded by
the Nullstellensatz". Our question is then, can we obtain a correspondence
whose closed ideals are exactly the radical ideals? To do so we will classify
the universal positive multiplicatively idempotent valuation semiring.

In the same way that we can construct a universal valuation semiring, or a
universal positive valuation semiring, we can construct a universal positive
multiplicatively idempotent semiring.

######  Definition 5.6.

The universal positive multiplicatively idempotent valuation semiring,
$\Gamma^{id}_{R}$, is the semiring

$\Gamma_{R}^{+}/\left\langle
x_{r}^{2}=x_{r}\right\rangle\cong\Gamma_{R}/\left\langle
x_{r}+1=1,x_{r}^{2}=x_{r}\right\rangle$

equipped with the valuation

$r\mapsto[x_{r}].$

######  Theorem 5.7.

This valuation is initial among the positive, multiplicatively idempotent
valuations of $R$.

Remark 5.8. When $I(R)$ is initial among positive valuations our initial
positive multiplicatively idempotent valuation semiring is the semiring
$I(R)/\left\langle(r)*(r)=(r)\right\rangle.$

######  Theorem 5.9.

The closed ideals of the correspondence between $I(R)$ and $\Gamma^{id}_{R}$
are exactly the radical congruences666In the case where $R$ is not Noetherian,
we look at the correspondence between $I(R)$ and the initial valuation among
the valuations whose codomains are complete semirings..

To prove this it will suffice to find a single valuation that distinguishes
between radicals. We will do that first and then present the proof of this
statement.

Example 5.10.  For a commutative ring $R$, we have a valuation $\tau:R\mapsto
T$ where $T$ is the closed Zariski topology on $\operatorname{Spec}(R)$ with

$\displaystyle A+_{T}B$ $\displaystyle=A\cap B,$ $\displaystyle A*_{T}B$
$\displaystyle=A\cup B,$ $\displaystyle 0_{T}$
$\displaystyle=\operatorname{Spec}(R),$ $\displaystyle 1_{T}$
$\displaystyle=\emptyset.$

$T$ is a complete idempotent semiring.

The associated valuation is the assignment of distinguished closed sets given
below.

$\tau:f\mapsto D_{f}=\\{P\text{ prime}:f\in P\\}$

Our correspondence is given by the maps:

$\displaystyle N:I(R)$ $\displaystyle\to T$ $\displaystyle N:J$
$\displaystyle\mapsto\bigcap_{j\in J}D_{j}=\\{P\text{ prime}:P\supseteq J\\}$
$\displaystyle I:T$ $\displaystyle\to I(R)$ $\displaystyle I:X$
$\displaystyle\mapsto\bigcap_{P\in X}P$

We can see that $IN(J)$ is the intersection of prime ideals containing $J$, so
$IN(J)=\sqrt{J}$.

We can now prove Theorem 5.9.

###### Proof.

We want to show that under the induced Galois correspondence between $I(R)$
and $\Gamma^{id}_{R}$ we have that $IN(J)=\sqrt{J}$. We know from Proposition
5.5 that $IN(J)\supseteq\sqrt{J}$, so it remains to show that
$IN(J)\subseteq\sqrt{J}$. To do so we will show that the closure given by the
valuation $\tau$ gives an upper bound to the closure on the universal
valuation.

Let $\tau:R\to T$ be the valuation in Example 5.9, and let $f$ be the unique
map from $\Gamma_{R}^{id}$ to $T$ that makes the following diagram commute:

${R}$${{\Gamma^{id}_{R}}}$${T}$$\scriptstyle{f}$$\scriptstyle{\nu}$$\scriptstyle{\tau}$

Let $N_{\Gamma}$ be the correspondence map $I(R)\to\Gamma^{id}_{R}$ and
$N_{T}$ be the correspondence map $I(R)\to T$. Similarly $I_{\Gamma}$ the map
$\Gamma^{id}_{R}\to I(R)$ and $I_{T}$ the map $T\to I(R)$

${R}$${{I(R)}}$${{\Gamma^{id}_{R}}}$${T}$$\scriptstyle{\nu}$$\scriptstyle{\tau}$$\scriptstyle{N_{\Gamma}}$$\scriptstyle{N_{T}}$$\scriptstyle{f}$

From the fact that $\tau=\nu\circ f$ and that $I_{T}(x)=\tau^{-1}(\\{t\in
T:t\geq x\\}$ we get the following statements.

$I_{T}(x)=\tau^{-1}(\\{t\in T:t\geq f(x)\\})=\nu^{-1}(f^{-1}(\\{t\in T:t\geq
f(x)))$ $\nu^{-1}(\\{\gamma\in\Gamma^{id}_{R}:\gamma\geq x\\})=I_{\Gamma}(x)$

We note that $f^{-1}(\\{t\in T:t\geq
f(x)\\})\supseteq\\{\gamma\in\Gamma^{id}_{R}:\gamma\geq x\\}$ and so we get:

$I_{T}(x)=\nu^{-1}(f^{-1}(\\{t\in T:t\geq
f(x)))\supseteq\nu^{-1}(\\{\gamma\in\Gamma^{id}_{R}:\gamma\geq
x\\})=I_{\Gamma}(x).$

Put succinctly, $I_{T}(f(x))\supseteq I_{\Gamma}(x)$

We note that as our $N_{\bullet}$ maps are unique maps from an initial object
we have that $N_{T}=fN_{\Gamma}$. Combined with the statement that
$I_{T}(f(x))\supseteq I_{\Gamma}(x)$ we get our upper bound.

$\sqrt{J}=IN_{T}(J)=I(f(N_{\Gamma}(J)))\supseteq IN_{\Gamma}(J)$

Together with Proposition 5.5 we get:

$\sqrt{J}\supseteq IN_{\Gamma}(J)\supseteq\sqrt{J}$ $IN_{\Gamma}(J)=\sqrt{J}$

∎

The valuation $\tau:R\to T$ can be used for more than bounding the closure
operator on the universal valuation. We can show that it is the universal
valuation.

######  Theorem 5.11.

The initial valuation among appropriately complete, multiplicatively
idempotent positive valuations is $\tau:R\to T$.

###### Proof.

We note that by definition $N:I(R)\to T$ is surjective, as each element in $T$
is a Zariski closed set, and so is of the form

$N(J)=\\{P\text{ prime}:P\supseteq J\\}$

for some ideal $J$. We also note that the map $N$ factors through
$I(R)/\left\langle x^{2}\sim x\right\rangle$, as $T$ is multiplicatively
idempotent.

We then need to show that if $N(J_{1})=N(J_{2})$ then $J_{1}\sim J_{2}$ under
the equivalence relation $x^{2}\sim x$, and we then get that the factored map
$I(R)/\left\langle x^{2}\sim x\right\rangle\to T$ is an isomorphism of
appropriately complete multiplicatively and additively idempotent semirings.

We first note that if $N(J_{1})=N(J_{2})$ then $\sqrt{J_{1}}=\sqrt{J_{2}}$.

Under the equivalence relation induced by $x^{2}\sim x$ we have that
$\sqrt{J}\sim J$.

$\displaystyle\sqrt{J}=\sum_{\\{r:r^{n}\in J\\}}(r)\sim\sum_{\\{r:r^{n}\in
J\\}}(r)^{n}=\sum_{r\in J}(r)=J$

If $N(J_{1})=N(J_{2})$ then $J_{1}\sim\sqrt{J_{1}}=\sqrt{J_{2}}\sim J_{2}$
under the congruence generated by $x^{2}\sim x$, and so the map
$I(R)/\left\langle x^{2}\sim x\right\rangle\to T$ is injective.

Thus we get that the factored map $\Gamma^{id}_{R}\cong I(R)/\left\langle
x^{2}=x\right\rangle\to T$ is both injective and surjective, and so it is an
isomorphism. ∎

Remark 5.12. Any assignment of lattice or set to elements of $R$ that “looks
like" assigning zeros to polynomials, “looks like" you are taking a quotient
of the coframe $\operatorname{Spec}(R)$, as that assignment factors through
$\operatorname{Spec}(R)$. While the zero locus itself is not necessarily the
universal valuation in this setting, we have that the Zariski topology on
$\operatorname{Spec}(R)$ is.

######  Corollary 5.13.

Let $D:R\to\operatorname{Spec}(R)$ be the map:

$r\mapsto\\{\mathfrak{p}:r\in\mathfrak{p}\\}$

As a distributive, inf-complete lattice (i.e. a coframe), the closed topology
$\operatorname{Spec}(R)$ with singleton closure given by $D$ is defined up to
isomorphism by the universal property:

For any positive, multiplicatively idempotent valuation $\nu:R\to\Gamma$,
there exists a unique map $\hat{\nu}:\operatorname{Spec}(R)\to\Gamma$ such
that $\nu=\hat{\nu}\circ D$.

We note that this universal property is the definition of the frame theoretic
spectrum of a ring given in section V.3.1 of [Joh82]. Through the generalized
definition of a valuation given by Jeffrey and Noah Giansiracusa in [GG16] we
can link this frame theoretic universal property directly to the field of
arithmetic valuation theory.

| Valuations |  | Positive (Complete)
---
Valuations
| Multiplicatively Idempotent
---
Positive (Complete)
Valuations
Universal Valuation |  $\mathbb{Z}$-submodules | The semiring of ideals |  | The coframe of the Zariski topology
---
on the prime spectrum
Figure 1: Initial valuations of a ring $R$, if $R$ is non-Noetherian then the
positive valuations are over complete semirings.

## References

  * [Ber23] William Bernardoni  “Applications of Generalized Universal Valuations”, 2023  arXiv:2305.08697 [math.RA]
  * [GG16] Jeffrey Giansiracusa and Noah Giansiracusa  “Equations of tropical varieties”  In _Duke Mathematical Journal_ 165.18 Duke University Press, 2016  DOI: 10.1215/00127094-3645544
  * [GG22] Jeffrey Giansiracusa and Noah Giansiracusa  “The universal tropicalization and the Berkovich analytification”, 2022  arXiv:1410.4348 [math.AG]
  * [Gol02] Martin Goldstern  “Completion of Semirings”, 2002  arXiv:math/0208134 [math.RA]
  * [Gol99] J.S. Golan  “Semirings and their Applications”  Springer Netherlands, 1999
  * [HOW06] Bernd Heidergott, Geert Jan Olsder and Jacob Woude  “Max Plus at Work: Modeling and Analysis of Synchronized Systems: A Course on Max-Plus Algebra and Its Applications”  Princeton University Press, 2006  URL: http://www.jstor.org/stable/j.ctt7zv8k3
  * [IMS07] I. Itenberg, G. Mikhalkin and E. Shustin  “Tropical Algebraic Geometry”, Oberwolfach Seminars Series  Birkhäuser, 2007
  * [Joh82] P.T. Johnstone  “Stone Spaces”, Cambridge Studies in Advanced Mathematics  Cambridge University Press, 1982
  * [Joy76] Andre Joyal  “Les théoremes de Chevalley-Tarski et remarques sur l’algèbre constructive”  In _Cah. Topologie Géom. Différ. Catégoriques_ 16, 1976, pp. 256–258
  * [JRT21] Jaiung Jun, Samarpita Ray and Jeffrey Tolliver  “Lattices, Spectral Spaces, and Closure Operations on Idempotent Semirings”, 2021  arXiv:2001.00808 [math.RA]
  * [Kru39] W. Krull  “Beiträge zur Arithmetik kommutativer Integritätsbereiche. VI. Der allgemeine Diskriminantensatz. Unverzweigte Ringerweiterungen”  In _Mathematische Zeitschrift_ 45.1, 1939, pp. 1–19  DOI: 10.1007/BF01580269
  * [Low] Zhen Lin Low  “Why is the spectrum of a commutative ring only concerned with fields?” URL:https://math.stackexchange.com/q/4305183 (version: 2021-11-13), Mathematics Stack Exchange  URL: https://math.stackexchange.com/q/4305183
  * [MS15] Diane Maclagan and Bernd Sturmfels  “Introduction to Tropical Geometry” 161, Graduate Studies in Mathematics  American Mathematical Society, Providence, RI, 2015
  * [nLa24] nLab authors  “adjoint functor theorem” Revision 78, https://ncatlab.org/nlab/show/adjoint+functor+theorem, 2024
  * [Tol16] Jeffrey Tolliver  “Extension of valuations in characteristic one”, 2016  arXiv:1605.06425 [math.RA]
  * [Vir06] Oleg Viro  “Patchworking real algebraic varieties”, 2006  arXiv:math/0611382 [math.AG]