# Zero-divisor graphs and zero-divisor functors

Enrico Sbarra  Enrico Sbarra - Dipartimento di Matematica - Università degli
Studi di Pisa - Largo Bruno Pontecorvo 5 - 56127 Pisa - Italy
<EMAIL_ADDRESS>and  Maurizio Zanardo  Maurizio Zanardo - Dipartimento
di Matematica - Università degli Studi di Pisa - Largo Bruno Pontecorvo 5 -
56127 Pisa - Italy<EMAIL_ADDRESS>
###### Abstract.

Inspired by a very recent work of A. Đurić, S. Jevđenić and N. Stopar, we
introduce a new definition of zero-divisor graphs attached to rings, that
includes all of the classical definitions already known in the literature. We
provide an interpretation of such graphs as images of a functor, that we call
zero-divisor functor and which is associated with a family of special
equivalence relations fixed beforehand. We thus recover and generalize many
known results for zero-divisor graphs and provide a framework which might be
useful for further investigations on this topic.

###### Key words and phrases:

Zero-divisor graph, zero-divisor functor, staircase graph, artinian rings

###### 2010 Mathematics Subject Classification:

Primary 05C25; Secondary 18A30, 13P25

## Introduction

Throughout this paper a ring will be a unitary commutative ring. The notion of
zero-divisor graph was introduced for the first time by Beck in 1988, see
[B88], with the purpose of investigating the structure of the underlying ring.
Given a ring $A$, Beck’s zero-divisor graph $G(A)$ is a graph whose vertex set
$V(A)$ is the set of all elements of $A$ and the set of edges $E(A)$ is
determined by the following zero-divisor relation on the elements of $A$: two
vertices $a,b\in V(A)=A$ are adjacent if and only if $a$ and $b$ satisfy the
relation $ab=0$ in $A$. The main concern in [B88] was to study the chromatic
and clique numbers of $G(A)$. Shortly after, in [AL99], Anderson and
Livingston introduced a slight modification of the definition of zero-divisor
graph, called $\Gamma(A)$, with the purpose of simplifying $G(A)$ and its
visualization without losing the relevant pieces of information therein
contained; their definition of the graph is considered as classical in the
literature. Many properties of $\Gamma(A)$, such as connectedness, see
[CSSS12], and its invariants, as for instance diameter, girth, and chromatic
number, have been later studied in [AM04] and [AL99]. These papers also
address the problem of finding rings $A$ whose zero-divisor graph belongs to
special classes of graphs, such as star graphs, complete graphs, $r$-partite
complete graphs, planar graphs etc., see also [AMY03].

Since $\Gamma(A)$ can have very many edges even when $A$ is finite of small
cardinality, in [M02] Mulay introduced a new relation, which we denote
$\asymp_{A}$, on the elements of $A$, defined as follows: given $a,b\in A$,
one lets $a\asymp_{A}b$ and, therefore, $a,b$ are adjacent as vertices of
$\Gamma(A)$, if and only if $\operatorname{Ann}(a)=\operatorname{Ann}(b)$. The
resulting graph, denoted by $\Gamma_{E}(A)$, has been later referred to as
compressed zero-divisor graph, see [AL12]. Thus, the purpose of
$\Gamma_{E}(A)$ is to simplify further the original graph by “compressing” it
or, in other words, by diminishing the “noise”, i.e. the redundancy, given by
those elements of $A$ which essentially behave in the same fashion. Compressed
zero-divisor graphs have been subsequently studied by several authors, cf. for
instance [SW11], [CSSS12] and [AL12].

In a very recent paper [DJS21], Đurić, Jevđenić and Stopar propose a new
approach to the study of zero-divisor graphs, by looking at functorial
properties they might be endowed with. They define a functor, called $\Theta$,
from the category of finite rings to that of graphs with loops; the image of a
ring $A$ is a compressed zero-divisor graph $\Theta(A)$, whose set of vertices
is given by $A/\sim_{A}$, where $a\sim_{A}b$ if and only if $(a)=(b)$ is the
same ideal of $A$, and where two equivalence classes $[a],[b]\in A/\sim_{A}$,
i.e. vertices of $\Theta(A)$, are adjacent if and only if $ab=0$ in $A$.

The starting point of our work is to develop one of the main ideas in [DJS21]
and the leitmotif of the first part of this paper is to search for equivalence
relations that are a good compromise between having functorial properties and
producing as little noise as possible on zero-divisor graphs, for instance
preserving their compression. From this perspective, one may wonder which are
the appropriate coarsest, i.e .minimal, equivalence relations such that the
associated zero-divisor functor satisfies a certain property. Another
interesting question is to study what happens if we introduce a degree of
freedom into the problem by letting each ring to be endowed with its own
equivalence relation.

We introduce some new definitions which provide the right framework to analyze
and, in some cases, to provide answers to such questions; we do so in the
central part of the paper which contains our main results. We have illustrated
an application of our approach in the last part of the paper, where we study
which rings $(A,\mathcal{R}_{A})$ have a finite associated zero-divisor graph
$\texttt{\large{Z}}(A,\mathcal{R}_{A})$. Since every artinian ring can be
written as finite product of local artinian rings, we choose as a good family
of equivalence relations $\\{\sim_{A}\\}$; in this way the associated functor
preserves finite products in both directions and this allows us to provide a
partial structure theorem for rings $A$ such that
$\texttt{\large{Z}}(A,\sim_{A})$ is finite.

After reviewing a few preliminaries on graphs in Section 1 and the classical
definitions of zero-divisor graphs in the literature in Section 2.1, we look
at which are the significant properties that an adjacency relation must have
to define a zero-divisor graph. We call such relations zero-divisor relations,
see Definition 2.2 and we observe in Proposition 2.4 that a zero-divisor
relation $\mathcal{R}_{A}$ is just an equivalence relation which is finer than
$\asymp_{A}$. In this way, given a ring $A$ and a zero-divisor relation
$\mathcal{R}_{A}$, we define a zero-divisor graph
$\texttt{\large{Z}}(A,\mathcal{R}_{A})$. Choosing the right $\mathcal{R}_{A}$,
by means of $\texttt{\large{Z}}(A,\mathcal{R}_{A})$ and of its strong
quotients we can recover the definitions of the zero-divisor graphs studied so
far. We prove in Theorem 2.6 that the compressed zero-divisor graphs
associated with $\mathcal{R}_{A}$, cf. Definition 2.3, are connected
independently of the chosen relation, generalizing [AL99, Theorem 2.3] and
[SW11, Proposition 1.4], see also [M02].

Next, we proceed to make Z into a functor from the category of commutative
rings CRing to that of graphs with loops Graph; we do this in Section 3.1. We
consider each object $A\in\textbf{CRing}$ associated with a fixed relation
$\mathcal{R}_{A}$, i.e. we fix a family $\\{\mathcal{R}_{A}\\}$ of equivalence
relations. We define first the key property we need on a family
$\\{\mathcal{R}_{A}\\}$ of zero-divisor relations, see Definition 3.1, and
provide some examples. We then define its associated zero-divisor functor
$\texttt{\large{Z}}\,:\,\textbf{CRing}\longrightarrow\textbf{Graph}$,
$(A,\mathcal{R}_{A})\mapsto\texttt{\large{Z}}(A,\mathcal{R}_{A})$, see
Definition 3.8.

We spend the rest of Section 3 in investigating the properties of the newly
defined functor Z. We first show under which conditions on
$\\{\mathcal{R}_{A}\\}$ the associated functor Z preserves binary products and
equalizers, see Propositions 3.11 and 3.15, and conclude that the families
$\\{=_{A}\\}$, $\\{\approx_{A}\\}$ and $\\{\sim_{A}\\}$ preserve finite
products, cf. Corollary 3.12, whereas $\\{=_{A}\\}$ is the only one which
preserves equalizers, and thus all finite limits. Corollary 3.12 also extends
to the case of infinite rings [DJS21, Proposition 3.4], see Corollary 3.13.
Section 3.4 contains one of the main results of this paper, Theorem 3.20 and
its consequence Corollary 3.22, which we call Inversion of Product. There we
show under which conditions on $\\{\mathcal{R}_{A}\\}$ the decomposition of
$\texttt{\large{Z}}(A,\mathcal{R}_{A})$ as a finite direct product of graphs
leads to a decomposition of $A$ as a finite direct product of rings. This
corrects and extends to the case of infinite rings the main result [DJS21,
Theorem 3.6].

Finally, in the last section, we apply the main results of Section 3 to study
the case when $\texttt{\large{Z}}(A)=\texttt{\large{Z}}(A,\mathcal{R}_{A})$ is
finite, focusing on the case $\mathcal{R}_{A}=\sim_{A}$, which turns out to be
of main interest, as we observe at the beginning of the section. We prove in
Theorem 4.1 that $\texttt{\large{Z}}(A)$ is finite exaclty when $A$ is the
direct product of an artinian PIR with a finite ring. Moreover, we prove that
local artinian PIRs are characterized as those rings $A$ for which
$\texttt{\large{Z}}(A)$ is a staircase graph. Finally, thanks to the Inversion
of Product Theorem, we characterize artinian PIRs as those rings whose
$\texttt{\large{Z}}(A)$ is a finite product of staircase graphs, see
Proposition 4.6 and Theorem 4.8, which extend to the case of infinite rings
the main results of [DJS21], Section 4 and 5.

##  1\. Preliminaries

All the rings we consider are unitary and commutative, and all rings
homomorphisms $\varphi\,:\,A\longrightarrow B$ map $1_{A}$ to $1_{B}$. We
denote by $\mathcal{I}_{A}$ the family of ideals of $A$ and by
$\mathcal{PI}_{A}\subseteq\mathcal{I}_{A}$ the subfamily of principal ideals.
By $\mathcal{D}(A)$ and $\mathcal{N}(A)$ we denote the set of zero-divisors
and the set of nilpotents of $A$, respectively. We also let
$\mathcal{D}^{*}(A):=\mathcal{D}(A)\smallsetminus\\{0\\}$. By $A^{\times}$ we
denote the set of invertible elements of $A$ and by $\operatorname{Ann}(a)$
the ideal $0:a$, with $a\in A$.

We recall next some definition and notation from Graph Theory, and we refer
the reader to [K11] for further details. A graph $G$ is an ordered couple
$(V(G),E(G))$, where $V=V(G)$ is a set, the vertex set of $G$ and $E=E(G)$ is
a binary symmetric relation on $V^{2}$, called the edge set of $G$. Given a
vertex $v\in V$, we call $N(v)=\\{w\in V\,:\,(v,w)\in E\\}$ the set of
adjacents of $v$; we call degree of $v$, and denote it by $\deg(v)$, the
cardinality of $N(v)$. We say that $v\in V$ has a loop if $v\in N(v)$ and,
when this is the case we call $(v,v)\in E$ a loop.

Let $G_{1}$ and $G_{2}$ be graphs. A graph (homo)morphism
$\varphi\,:\,G_{1}\rightarrow G_{2}$ is a map $V(G_{1})\to V(G_{2})$ such that
$\big{(}(x,y)\in E(G_{1})\Rightarrow(\varphi(x),\varphi(y))\in E(G_{2})$, for
all $x,y\in V(G_{1})\big{)}$. Similarly, a graph comorphism
$\varphi:G_{1}\rightarrow G_{2}$ is a map $V(G_{1})\to V(G_{2})$ such that

$\big{(}(\varphi(x),\varphi(y))\in E(G_{2})\Rightarrow(x,y)\in E(G_{1})$, for
all $x,y\in V(G_{1})\big{)}$. When $\varphi$ is both a graph morphism and
comorphism $\varphi$ is called strong or full morphism. We call a graph
morphism $\varphi$ (graph) isomorphism if it bijective and its inverse is also
a graph morphism. Observe that $\varphi$ is an isomorphism if and only if it
is a bijective strong morphism.

Given two graphs $H,G$ with $V(H)\subseteq V(G)$, we say that $H$ is a
subgraph of $G$ if the inclusion map is a graph morphism, and a subgraph $H$
of $G$ is called induced if the inclusion map is a strong morphism. Let now
$\mathcal{R}$ be an equivalence relation on $V=V(G)$; we call the quotient
graph of $G$ by $\mathcal{R}$, and we denote it by $G/\mathcal{R}$, the graph
whose vertex set is $V/\mathcal{R}$ and, given $[x],[y]\in V(G/\mathcal{R})$,
one has $([x],[y])\in E(G/\mathcal{R})$ if and only there exist
$x^{\prime}\in[x]$ and $y^{\prime}\in[y]$ with $(x^{\prime},y^{\prime})\in
E(G)$, i.e. $\pi_{\mathcal{R}}\,:\,G\longrightarrow G/\mathcal{R}$ defined by
$v\mapsto[v]$ is a graph morphism. When $\pi_{\mathcal{R}}$ is a strong
morphism, we say that $G/\mathcal{R}$ is a _strong quotient_ of $G$.

We conclude this section by recalling some common operations defined on
graphs. The _Kronecker product_ , or simply _product_ , of $G_{1}$ and
$G_{2}$, denoted by $G_{1}\times G_{2},$ is the graph whose vertex set is
$V(G_{1})\times V(G_{2})$ and two couple of vertices are adjacent if and only
if both the first and second components of the couples are adjacent. The
_coproduct_ $G_{1}\sqcup G_{2}$ of two graphs $G_{1}$ and $G_{2}$ is the graph
whose vertex set is the disjoint union of the vertices of $G_{1}$ and $G_{2}$
and two vertices are adjacent if and only if they are adjacent in $G_{1}$ or
in $G_{2}$, i.e. its edge set is the disjoint union of the edges of both
graphs. Finally, given two graph morphisms $\phi,\psi:G_{1}\to G_{2}$, the
_equalizer_ $\operatorname{Eq}(\varphi,\psi)$ is the (strong) subgraph induced
by the subset of vertex where $\varphi$ and $\psi$ coincide.

The pair of data given by the class of all graphs and graph morphisms, with
composition and identities defined in the natural way, is a category, that
will be denoted by Graph, with product, coproduct and equalizer defined as
above, cf. [K11, Sections 4.1 and 4.2].

##  2\. Zero-divisor graphs

There are various definitions of zero-divisor graphs attached to commutative
rings which appear in the literature and we are going to review next some of
the most significative. As we did above for graphs, given a ring $A$ and an
equivalence relation $\mathcal{R}$ on $A$, we shall denote by $[a]$ the
equivalence class of $a$ with respect to $\mathcal{R}$. Two elements $a,b\in
A$ are called _associated_ , and we write $a\sim b$, if $(a)=(b)$. We say that
$a,b\in A$ are _strongly associated_ , and we write $a\approx b$, if there
exists $u\in A^{\times}$ such that $a=ub$. Finally, we call $a,b\in A$
_equiannihilated_ , and we write $a\asymp b$, if
$\operatorname{Ann}(a)=\operatorname{Ann}(b)$. It is immediate to see that
$\sim,\approx$ and $\asymp$ are indeed equivalence relations.

###  2.1. Definitions in the literature

Essentially, a zero-divisor graph attached to a ring $A$ is a graph whose
vertex set is $A/\mathcal{R}_{A}$, where $\mathcal{R}_{A}$ is an equivalence
relation on $A$, together with an adjacency relation.

In [B88], $\mathcal{R}_{A}$ is the trivial relation, i.e. equality; therefore,
Beck’s zero-divisor graph $G(A)$ has vertex set $V(G(A))=A$. Loops are not
allowed in $G(A)$; in fact, the adjacency relation $E=E(G(A))$ is given by

$(a,b)\in E\text{\,\,if and only if\,\,}a\neq b\,\,\text{and}\,\,ab=0.$

In [AL99], the vertex set is just $\mathcal{D}^{*}(A)$ and the resulting graph
$\Gamma(A)$ is the induced subgraph of $G(A)$ by $\mathcal{D}^{*}(A).$

In [M02] the so-called compressed zero-divisor graph $\Gamma_{E}(A)$ has
vertex set $\mathcal{D}^{*}(A)/\asymp$, and the adjacency relation $E$ is
defined as

$\big{(}[a],[b]\big{)}\in E\,\,\text{if and only if\,\,}a\neq
b\,\,\text{and}\,\,ab=0;$

it is a graph with no loop which is a strong quotient of $\Gamma(A)$.

In [DJS21], functorial properties of the compressed zero-divisor graph have
been studied in the case of a finite ring $A$, by means of a functor $\Theta$;
the image $\Theta(A)$ results to be a compressed zero-divisor-graph whose
vertex set is $A/\\!\approx$ and the adjacency relations $E$ is defined by

$\big{(}[a],[b]\big{)}\in E\,\,\text{if and only if}\,\,ab=0.$

At the end of the same paper, the authors propose a modification, apt to treat
the general case of infinite rings.

######  Definition 2.1 ([DJS21], Definition 6.1).

Let $A$ be a ring. The _zero-divisor graph on principal ideals_ $\Theta(A)$
has vertex set $\mathcal{PI}_{A}$ and the adjacency relation $E$ is defined by

$\big{(}(a),(b)\big{)}\in E\,\,\text{if and only if}\,\,ab=0,$

i.e. two principal ideals are adjacent if the product of their principal
generators is zero or, equivalently if their product is the zero ideal.

With a slight abuse of notation, we shall use both $(a)$ and $[a]$ to denote a
vertex of $\Theta(A)$, since $(a)\mapsto[a]$ is a 1:1 correspondence between
$\mathcal{PI}_{A}$ and $A/\sim.$

Unlike in the other graphs introduced so far, loops are allowed in
$\Theta(A)$, and this fact is essential in order to provide categorical
properties of the functor $\Theta$ for both finite and infinite rings.

###  2.2. Zero-divisor relations and a new definition

One of the key observations in order to provide a generalization is that all
of the adjacency relations which have been considered so far all share the
same “minimal” property, which lead us to the following definition.

######  Definition 2.2.

An equivalence relation $\mathcal{R}$ on $A$ is called _zero-divisor relation_
if, taken any $a\mathcal{R}a^{\prime}$ e $b\mathcal{R}b^{\prime}$, it holds
that $\left(ab=0\Rightarrow a^{\prime}b^{\prime}=0\right)$.

Observe that an equivalence relation $\mathcal{R}$ is zero-divisor if and only
if the adjacency relation $E$ on $A/\mathcal{R}$, defined by $([a],[b])\in
E\\!\iff\\!ab=0$, is well defined. This fact suggests the choice of the name
and puts us in a position to provide the announced definition.

######  Definition 2.3.

A _zero-divisor graph on $A$_ is a graph whose vertex set is $A/\mathcal{R}$,
for some zero-divisor relation $\mathcal{R}$ on $A$, and $[a]$ and $[b]$ are
adjacent if and only if $ab=0$; we denote it by
$\texttt{\large{Z}}(A,\mathcal{R})$.

Given any subset $X\subset A$, we also let $\texttt{\large{Z}}(X,\mathcal{R})$
be the subgraph of $\texttt{\large{Z}}(A,\mathcal{R})$ induced by $\\{[a]\mid
a\in X\\}$.

When there is no ambiguity, we shall denote
$\texttt{\large{Z}}(A,\mathcal{R})$, resp.
$\texttt{\large{Z}}(X,\mathcal{R})$, simply by $\texttt{\large{Z}}(A)$, resp.
$\texttt{\large{Z}}(X)$. A case of special interest is when
$X=\mathcal{D}^{*}(A)$, since the essential part of a zero-divisor graph is
given by the equivalence classes of the elements of $X$.

By the very definitions, we can capture the graphs $G(A)$, $\Gamma(A)$,
$\Gamma_{E}(A)$ by means of $\texttt{\large{Z}}(A,=)$,
$\texttt{\large{Z}}(D^{*}(A),=)$, $\texttt{\large{Z}}(D^{*}(A),\asymp)$
respectively, removing their loops. Moreover, for any finite ring $A$, we have
$\Theta(A)=\texttt{\large{Z}}(A,\approx)$

Among all zero-divisor relations on $A$, the relation $\asymp$ turns out to be
the coarsest, and this somewhat suggests the appellative compressed of the
graph $\Gamma_{E}(A)$. Also, maybe surprisingly, the viceversa holds, i.e. if
$\mathcal{R}$ is finer than the relation $\asymp$, then it is a zero-divisor
relation. We record this fact in the next result.

######  Proposition 2.4.

Let $\mathcal{R}$ be a an equivalence relation on $A$; then $\mathcal{R}$ is
zero-divisor if and only if $\mathcal{R}$ is finer than $\asymp$, i.e. for all
$a$ and $b$, $a\mathcal{R}b$ implies $a\asymp b$.

###### Proof.

Let us take $a\mathcal{R}b$, for a given zero-divisor relation $\mathcal{R}$
on $A$ and, by symmetry, it is sufficient to prove that
$\operatorname{Ann}(a)\subseteq\operatorname{Ann}(b)$. Given any
$c\in\operatorname{Ann}(a)$, we have that $ca=0$ and, since $a\mathcal{R}b$
and $c\mathcal{R}c$, it follows that $cb=0$.
Viceversa, if $a\mathcal{R}a^{\prime}$ and $b\mathcal{R}b^{\prime}$ for a
finer relation than $\asymp$ then
$\operatorname{Ann}(a)=\operatorname{Ann}(a^{\prime})$ and
$\operatorname{Ann}(b)=\operatorname{Ann}(b^{\prime})$. So, if $ab=0$, then
$b\in\operatorname{Ann}(a)=\operatorname{Ann}(a^{\prime})$, i.e.
$a^{\prime}b=0$; therefore
$a^{\prime}\in\operatorname{Ann}(b)=\operatorname{Ann}(b^{\prime})$, i.e.
$a^{\prime}b^{\prime}=0$. ∎

A consequence of the previous proposition is that, for all zero-divisor
relations on $A$, it holds that $[0]=\\{0\\}$, because
$A=\operatorname{Ann}(0)=\operatorname{Ann}(a)$ implies $a=1\cdot a=0$. It is
also easy to see that $=$, $\approx$ and $\sim$ are zero-divisor relations,
because $a=a^{\prime}$ implies $a\approx a^{\prime}$, which in turn yields
$(a)=(a^{\prime})$ and, thus,
$\operatorname{Ann}(a)=\operatorname{Ann}(a^{\prime})$ for all
$a,a^{\prime}\in A.$
Also observe that, in general, $\sim$ is not the same as $\asymp$; take for
instance the ring ${\mathbb{Z}}$, where we have $2\asymp 1$ but
$(2)\neq(1)={\mathbb{Z}}.$

######  Remark 2.5.

Observe that, given zero-divisor relations $\mathcal{R},\mathcal{R}^{\prime}$
on $A$ such that $\mathcal{R}$ is finer than $\mathcal{R}^{\prime}$, then
$\texttt{\large{Z}}(A,\mathcal{R}^{\prime})$ is a strong quotient of
$\texttt{\large{Z}}(A,\mathcal{R})$.

Many known facts about zero-divisor graphs can be generalized in terms of Z,
as for instance the following, cf. [AL99] Theorem 2.3, [M02], [SW11]
Proposition 1.4; see also [SZ22] for more results of this type.

######  Theorem 2.6.

Let $\mathcal{R}$ be a zero-divisor relation on $A$; then the subgraph
$\texttt{\large{Z}}(\mathcal{D}^{*}(A),\mathcal{R})$, when not empty, is
connected with diameter $\leqslant 3$.

###### Proof.

Consider the graph $G$ which has the same vertex set as
$\texttt{\large{Z}}(\mathcal{D}^{*}(A),\mathcal{R})$ and with edge set
obtained by $E(\texttt{\large{Z}}(\mathcal{D}^{*}(A),\mathcal{R}))$ by
removing all the loops. Observe that, without loss of generality we can prove
the statement for $G$, since adding loops does not ruin connectedness nor
affects the diameter. Since $G$ is a strong quotient of $\Gamma$ by Remark
2.5, the conclusion follows from [AL99], Theorem 2.3. ∎

##  3\. Zero-divisor functors

We denote by CRing the category of commutative rings and we recall that the
categorical product in CRing is the direct product of rings, whereas the
categorical coproduct is given by the underlying tensor product of abelian
groups, equipped with its canonically induced commutative ring structure.

In this section we generalize to infinite rings the ideas of [DJS21], and
prove some properties of the map
$(A,\mathcal{R})\mapsto\texttt{\large{Z}}(A,\mathcal{R})$ that make it a
functor from the category CRing to the category Graph, by fixing beforehand a
family of equivalence relations indexed by the class of all rings. Some
conditions on the family of relations we consider are needed, for Z to behave
well on ring homomorphisms.

###  3.1. Functorial families of equivalence relations

Let now $\\{\mathcal{R_{A}}\\}$, where $A$ varies in CRing, denote a
collection of equivalence relations.

######  Definition 3.1.

We say that $\\{\mathcal{R_{A}}\\}$ is a _functorial family_ in CRing if, for
all morphisms $f\,:\,A\rightarrow B$ in CRing, it holds that
$a\mathcal{R}_{A}a^{\prime}\Rightarrow f(a)\mathcal{R}_{B}f(a^{\prime})$.

A collection of equivalence relations is thus functorial in CRing if every
ring homomorphism maps equivalence classes to equivalence classes. The
appellative functorial in the previous definition is motivated by the
following fact.

######  Proposition 3.2.

Let $\\{\mathcal{R}_{A}\\}$ be a family of equivalence relations; then
$\\{\mathcal{R}_{A}\\}$ is functorial in CRing if and only if there exists a
functor $F:\textbf{CRing}\rightarrow\textbf{Set}$ such that
$F(A)=A/\mathcal{R}_{A}$ and $F(f)([a])=[f(a)]$ for all rings $A$ and
homomorphisms $f\,:\,A\rightarrow B$.

###### Proof.

The map $F(f)$ is well-defined if and only if $\\{\mathcal{R}_{A}\\}$ is
functorial. The equalities $F(\operatorname{id}_{A})=\operatorname{id}_{F(A)}$
and $F(f\circ g)=F(f)\circ F(g)$ are a direct consequence of the definition of
$F(f)$. ∎

######  Example 3.3.

The following are examples of functorial families of relations.

  1. 1.

$\\{\mathcal{R}_{A}\\}=\\{=_{A}\\}$, the _trivial_ functorial family;

  2. 2.

$\\{\mathcal{R}_{A}\\}=\\{\approx_{A}\\}$, since, if $a_{1}=ua_{2}$ where
$u\in A^{\times}$, then $f(a_{1})=f(u)f(a_{2})$ and $f(u)$ is a unit for every
ring homomorphism $f$;

  3. 3.

$\\{\mathcal{R}_{A}\\}=\\{\sim_{A}\\}$, since, if $(a_{1})=(a_{2})$ then
$(f(a_{1}))=(f(a_{2}))$ holds, for all homomorphisms $f$.

######  Example 3.4.

1\. Let $\\{\mathcal{R}_{A}\\}$ be a family of equivalence relations such that
$A/\mathcal{R}_{A}$ has only one equivalence class for each ring $A$; then
$\\{\mathcal{R}_{A}\\}$ is a functorial family, which we call _totally
degenerate_.

2\. The family $\\{\mathcal{R}_{A}\\}=\\{\asymp_{A}\\}$ is not functorial. To
this end, take the projection $\pi\,:\,{\mathbb{Z}}\to{\mathbb{Z}}/(2)$,
$n\mapsto\bar{n}$; we then have
$\operatorname{Ann}(\bar{1})=\operatorname{Ann}(\bar{2})=0$ but
$\operatorname{Ann}(\bar{1})=0\neq{\mathbb{Z}}/(2)=\operatorname{Ann}(\bar{2})$.

All the families of Example 3.3 are in a sense tame, since they consist
essentially of the same relation defined on all of the rings. In general,
things can be more intricated, as in the following example.

######  Example 3.5.

Consider $\\{\mathcal{R}_{A}\\}=\\{\sim_{A}\\}$ and
$\\{\mathcal{R}^{\prime}_{A}\\}=\\{\approx_{A}\\}$, which are functorial as we
have seen in Example 3.3 and notice that, for all $A$, we have
$\mathcal{R}^{\prime}_{A}$ is finer than $\mathcal{R}_{A}$. Also observe that
the set of nilpotents $\mathcal{N}(A)$ can be partioned as
$\mathcal{N}(A)=\bigsqcup_{a\in\mathcal{N}(A)}[a]_{\mathcal{R}_{A}}$. We
define now the family $\\{\mathcal{R}^{\prime\prime}_{A}\\}$ as follows: for
all $A\in\textbf{CRing}$, $a,a^{\prime}\in A$ we let

$a\mathcal{R}^{\prime\prime}_{A}a^{\prime}\text{ if and only if
}\big{(}a\mathcal{R}_{A}a^{\prime}\text{ when
}a,a^{\prime}\in\mathcal{N}(A)\text{ and
}a\mathcal{R}^{\prime}_{A}a^{\prime}\text{ otherwise}\big{)}.$

Since $\\{\mathcal{R}^{\prime}_{A}\\}$ is finer than $\\{\mathcal{R}_{A}\\}$
and all homomorphism $f:A\to B$ are such that
$f(\mathcal{N}(A))\subseteq\mathcal{N}(B)$, the family
$\\{\mathcal{R}^{\prime\prime}_{A}\\}$ is a functorial family too.

We say that a family of equivalence relations $\\{\mathcal{R}_{A}\\}$ is
_degenerate_ if there exists a ring $A$ such that $[0]\neq\\{0\\}$ in
$A/\mathcal{R}_{A}$ and non degenerate otherwise.

######  Proposition 3.6.

Every non degenerate functorial family of equivalence relations
$\\{\mathcal{R}_{A}\\}$ is such that $\big{(}a\mathcal{R}_{A}b\Rightarrow
a\sim_{A}b$, for all $A\in\textbf{CRing}\big{)}$, i.e. each $\mathcal{R}_{A}$
is finer that $\sim_{A}$.

###### Proof.

For every ring $A$ and elements $a,b\in A$, consider the projection of $A$
onto $A/(a)$; since $\\{\mathcal{R}_{A}\\}$ is functorial we have
$\bar{a}\mathcal{R}_{A/(a)}\bar{b}$ and $\bar{a}=\bar{0}$; thus, from
$[0]=\\{0\\}$ we get that $(b)\subseteq(a)$. By symmetry we also obtain
$(a)\subseteq(b)$, as desired. ∎

######  Remark 3.7.

Unlike the case of zero-divisor relations, see Proposition 2.4, the converse
of the above proposition does not hold. Take for istance the family
$\\{\mathcal{R}_{A}\\}=\\{=_{A}:A\in\textbf{CRing}\smallsetminus{\mathbb{Z}}\\}\cup\\{\sim_{\mathbb{Z}}\\}$
and the inclusion homomorphism ${\mathbb{Z}}\subseteq{\mathbb{Q}}$. Since
$1\sim_{\mathbb{Z}}\\!-\\!1$ and $1\neq_{\mathbb{Q}}\\!-\\!1$, we may conclude
that $\\{\mathcal{R}_{A}\\}$ is not functorial.

###  3.2. Definition of zero-divisor functors

With the above definitions, by taking a _zero-divisor functorial family_
$\\{\mathcal{R}_{A}\\}$, i.e. a functorial family such that $\mathcal{R}_{A}$
is a zero-divisor relation for every $A\in\textbf{CRing}$, we are now in a
position to define a functor Z that generalizes $\Theta$. Notice that all the
functorial families in the Example 3.3 are zero-divisor.

######  Proposition 3.8.

Let $\\{\mathcal{R}_{A}\\}$ be a zero-divisor functorial family in CRing;
then, there exists a functor
$\texttt{\large{Z}}\,:\,\textbf{CRing}\rightarrow\textbf{Graph}$ such that
$\texttt{\large{Z}}(A)$ is the zero-divisor graph
$\texttt{\large{Z}}(A,\mathcal{R}_{A})$ and
$\texttt{\large{Z}}(f)([a])=[f(a)]$ for all $f\,:\,A\rightarrow B$.
We call Z the _zero-divisor functor associated to $\\{\mathcal{R}_{A}\\}$_.

###### Proof.

Since $\mathcal{R}_{A}$ is a zero-divisor relation, $\texttt{\large{Z}}(A)$ is
a zero-divisor graph and, since $\\{\mathcal{R}_{A}\\}$ is functorial, by
Proposition 3.2 the map
$\texttt{\large{Z}}(f):V(\texttt{\large{Z}}(A))\rightarrow
V(\texttt{\large{Z}}(B))$ is well-defined. Finally, it is easy to verify that
$\texttt{\large{Z}}(f)$ is a graph morphism, because $ab=0$ implies
$0=f(ab)=f(a)f(b)$. ∎

The functor Z we just defined by means of Proposition 3.8 generalizes the
functor $\Theta$ defined in [DJS21, Definition 3.2], over the category of
finite rings, and [DJS21, Definition 6.1], see also Definition 2.1. In fact,
if we choose $\\{\mathcal{R}_{A}\\}=\\{\sim_{A}\\}$, we will have that
$\texttt{\large{Z}}=\Theta$.

######  Remark 3.9.

By Proposition 2.4, a zero-divisor family $\\{\mathcal{R}_{A}\\}$ is non
degenerate, since $[0]=\\{0\\}$ when $\mathcal{R}_{A}=\asymp_{A}$. Thus, by
Proposition 3.6, all zero-divisor functorial families range between
$\\{=_{A}\\}$, which is the finest, and $\\{\sim_{A}\\}$, which is the
coarsest. In particular, the family $\\{\sim_{A}\\}$ is the one that
compresses the most each of the zero-divisor graphs $\texttt{\large{Z}}(A)$.

######  Example 3.10.

Clearly, if $f$ is an isomorphism in CRing, then $\texttt{\large{Z}}(f)$ is an
isomorphism in Graph; therefore the rings $A={\mathbb{Z}}/(4)[x]/(2x,x^{2})$
and $B={\mathbb{Z}}/(4)[x]/(2x,x^{2}-2)$ are not isomorphic, since
$\texttt{\large{Z}}(A,\sim_{A})$ has five elements and
$\texttt{\large{Z}}(B,\sim_{B})$ has four.

Recall that $f\,:\,X\to Y$ is a monomorphism in a category C when for every
object $Z$ and every pair of morphisms $g_{1},g_{2}\,:\,A\to X$ it holds that
$\big{(}(fg_{1}=fg_{2})\Rightarrow(g_{1}=g_{2})\big{)}$; similarly, $f$ is an
epimorphism in C when for every object $Z$ and every pair of morphisms
$g_{1},g_{2}\,:\,Y\to Z$ then
$\big{(}(g_{1}f=g_{2}f)\Rightarrow(g_{1}=g_{2})\big{)}$. In general, if $f$ is
a monomorphism, resp. an epimorphism, in CRing then, $\texttt{\large{Z}}(f)$
is not a monomorphism, resp. an epimorphism of graphs. For instance, consider
again as $f$ the inclusion of ${\mathbb{Z}}$ in ${\mathbb{Q}}$ which is mono
and epi in CRing and take the functorial family $\\{\sim_{A}\\}$. Then
$\texttt{\large{Z}}\,:\,\texttt{\large{Z}}({\mathbb{Z}},\sim)\longrightarrow\texttt{\large{Z}}({\mathbb{Q}},\sim)$
is a morphism from an infinite graph to a graph with two vertices, which is
not a monomorphism in Graph. Instead, if we take the family $\\{=_{A}\\}$,
then the morphism
$\texttt{\large{Z}}(f)\,:\,\texttt{\large{Z}}({\mathbb{Z}},=)\longrightarrow\texttt{\large{Z}}({\mathbb{Q}},=)$
is not an epimorphism in Graph. Furthermore, note that in general Z does not
reflects isomorphisms in CRing and, thus, neither mono nor epimorphisms. For
instance, consider the family $\\{\sim_{A}\\}$ and the inclusion
${\mathbb{Q}}\subseteq{\mathbb{Q}}(\sqrt{2})$. By applying Z we obtain an
isomorphism of graphs, since ${\mathbb{Q}}$ and ${\mathbb{Q}}(\sqrt{2})$ are
both fields, but ${\mathbb{Q}}(\sqrt{2})\not\simeq{\mathbb{Q}}$ in CRing.

We conclude this part by observing that, in general, the functor Z is neither
faithful nor full. Consider the family $\\{\sim_{A}\\}$ and the identity and
conjugation automorphisms of ${\mathbb{C}}$. Then, their images by Z are
equal; also, not all graphs morphisms
$\texttt{\large{Z}}({\mathbb{C}})\longrightarrow\texttt{\large{Z}}({\mathbb{C}})$
are images of rings morphisms, for there is the trivial graph morphism
$\texttt{\large{Z}}({\mathbb{C}})\to\texttt{\large{Z}}({\mathbb{C}})$ that
maps to 0 but, on the other hand, no ring homomorphism
$f\,:\,{\mathbb{C}}\longrightarrow{\mathbb{C}}$ is such that $f(1)=0$.

###  3.3. Finite products and finite limits. Coproducts.

Let us fix a zero-divisor functorial family $\\{\mathcal{R}_{A}\\}$. We are
going to study next under which conditions its associated zero-divisor functor
Z preserves finite limits. To this end, it is sufficient to understand when Z
preserves the terminal object, binary products and equalizers, cf. for
instance [B94], Proposition 2.8.2.

Clearly, independently of the choice of the family,
$\texttt{\large{Z}}(\textbf{0})$ is isomorphic to the graph composed by a
single vertex with a loop, which is the terminal object in Graph.

Consider now $A,B\in\textbf{CRing}$, the projections $A\times B\to A$,
$A\times B\to B$ and the induced graph morphisms $\texttt{\large{Z}}(A\times
B)\to\texttt{\large{Z}}(A)$, $\texttt{\large{Z}}(A\times
B)\to\texttt{\large{Z}}(B)$. By the universal property of the product in
Graph, there exists an unique canonical graph morphism
$\phi\,:\,\texttt{\large{Z}}(A\times
B)\to\texttt{\large{Z}}(A)\times\texttt{\large{Z}}(B)$.

######  Proposition 3.11.

Let $A,B$ be rings; then

  1. 1.

$\phi\,:\,\texttt{\large{Z}}(A\times
B)\longrightarrow\texttt{\large{Z}}(A)\times\texttt{\large{Z}}(B)$ is a strong
graph epimorphism;

  2. 2.

if $\\{\mathcal{R}_{A}\\}$ also satisfies the following property

$a\mathcal{R}_{A}a^{\prime},b\mathcal{R}_{B}b^{\prime}\Longrightarrow(a,b)\mathcal{R}_{A\times
B}(a^{\prime},b^{\prime})\text{\, for all\,}a,a^{\prime}\in A\text{\,
and\,}b,b^{\prime}\in B,$

for all $A,B$ in CRing, then $\phi$ is an isomorphism.

###### Proof.

1\. By the very definition, $\phi[(a,b)]=([a],[b])$. Observe that $[(a,b)]$
and $[(a^{\prime},b^{\prime})]$ are adjacent in $\texttt{\large{Z}}(A\times
B)$ if and only if $(a,b)(a^{\prime},b^{\prime})=(aa^{\prime},bb^{\prime})=0$,
if and only if $([a],[a^{\prime}])\in E(\texttt{\large{Z}}(A))$ and
$([b],[b^{\prime}])\in E(\texttt{\large{Z}}(B))$; thus, $\phi$ is a strong
morphism of graphs, which is evidently surjective.

2\. It is immediate. ∎

Notice that, since a composition of strong graph epimorphisms, resp.
isomorphisms, is a strong graph epimorphism, resp. isomorphism, the previous
proposition extends to arbitrary finite products.

######  Corollary 3.12.

Let $\\{\mathcal{R}_{A}\\}$ be one of the zero-divisor functorial families
$\\{=_{A}\\}$, $\\{\approx_{A}\\}$, or $\\{\sim_{A}\\}$. Then, for all
$A_{1},\dots,A_{n}\in\textbf{CRing}$ we have
$\texttt{\large{Z}}(A_{1}\times\cdots\times
A_{n})\simeq\texttt{\large{Z}}(A_{1})\times\cdots\times\texttt{\large{Z}}(A_{n})$.

###### Proof.

One only needs to show that $\\{\approx_{A}\\}$ and $\\{\sim_{A}\\}$ satisfy
the hypothesis of Proposition 3.11.2. Let $a\sim_{A}a^{\prime}$ and
$b\sim_{B}b^{\prime}$; then there exist $h,h^{\prime}\in A$ and
$k,k^{\prime}\in B$ such that $a=ha^{\prime}$, $a^{\prime}=h^{\prime}a$,
$b=kb^{\prime}$, $b^{\prime}=k^{\prime}b$. Thus,
$(a,b)=(h,k)(a^{\prime},b^{\prime})$ and
$(a^{\prime},b^{\prime})=(h^{\prime},k^{\prime})(a,b)$, i.e.
$(a,b)\sim_{A\times B}(a^{\prime},b^{\prime})$.

The proof for $\\{\approx_{A}\\}$ is utterly similar. ∎

Applying the previous Corollary to the family
$\\{\mathcal{R}_{A}\\}=\\{\approx_{A}\\}$ in the case of finite rings we
recover the following result.

######  Corollary 3.13 ([DJS21], Proposition 3.4).

The functor $\Theta\,:\,\textbf{FinCRing}\longrightarrow\textbf{Graph}$
preserves finite products.

$\texttt{\large{Z}}({\mathbb{Z}}/(2),\sim)$

######  Example 3.14.

Proceeding like in Example 3.5, consider the functorial families
$\\{\mathcal{R}_{A}\\}=\\{\sim_{A}\\}$ and
$\\{\mathcal{R}^{\prime}_{A}\\}=\\{=_{A}\\}$; notice that
$A^{\times}=\bigsqcup_{a\in A^{\times}}[a]_{\mathcal{R}_{A}}$, and define
$\mathcal{R}^{\prime\prime}_{A}$ by means of $\\{\mathcal{R}_{A}\\}$ and
$\\{\mathcal{R}^{\prime}_{A}\\}$. In the proof of Corollary 3.12 we have seen
that $\\{\mathcal{R}_{A}\\}$ and $\\{\mathcal{R}^{\prime}_{A}\\}$ verify the
hypothesis of Proposition 3.11.2, nonetheless
$\\{\mathcal{R}^{\prime\prime}_{A}\\}$ does not. Take for instance
$A_{1}=A_{2}={\mathbb{Z}}$; then
$1\mathcal{R}^{\prime\prime}_{\mathbb{Z}}\small{-}1$,
$2\mathcal{R}^{\prime\prime}_{\mathbb{Z}}2$, but it does not hold that
$(1,2)\mathcal{R}^{\prime\prime}_{{\mathbb{Z}}\times{\mathbb{Z}}}(\small{-}1,2)$.
Applying the associated zero-divisor functor Z we find out that
$\texttt{\large{Z}}({\mathbb{Z}}/(2)\times{\mathbb{Z}}/(4))$ has seven
vertices, whereas
$\texttt{\large{Z}}({\mathbb{Z}}/(2))\times\texttt{\large{Z}}({\mathbb{Z}}/(4))$
has six and, hence, they are not isomorphic.

We observe that, in general, Z does not reflects direct products. Consider for
instance two morphisms
${\mathbb{Q}}\times{\mathbb{Z}}/(2)\longrightarrow{\mathbb{Q}}(\sqrt{2})$ and
${\mathbb{Q}}\times{\mathbb{Z}}/(2)\longrightarrow{\mathbb{Z}}/(2)$; then
$\texttt{\large{Z}}({\mathbb{Q}}(\sqrt{2})\times{\mathbb{Z}}/(2),\sim)\simeq\texttt{\large{Z}}({\mathbb{Q}}(\sqrt{2}),\sim)\times\texttt{\large{Z}}({\mathbb{Z}}/(2),\sim)\simeq\texttt{\large{Z}}({\mathbb{Q}},\sim)\times\texttt{\large{Z}}({\mathbb{Z}}/(2),\sim)$
in Graph, whereas
${\mathbb{Q}}\times{\mathbb{Z}}/(2)\not\simeq{\mathbb{Q}}(\sqrt{2})\times{\mathbb{Z}}/(2)$
in CRing.

Now we look at equalizers. Consider $A,B\in\textbf{CRing}$ and $f,g\,:\,A\to
B$ morphisms in CRing. Recall that the _equalizer_ $\operatorname{Eq}(f,g)$ is
the subring $\ker(f-g)\subseteq A$. Therefore, we have an inclusion map
$\operatorname{Eq}(f,g)\to A$ that induces a morphism
$\texttt{\large{Z}}(\operatorname{Eq}(f,g))\to\texttt{\large{Z}}(A)$ in Graph,
which composed by $\texttt{\large{Z}}(f)$ or by $\texttt{\large{Z}}(g)$
returns the same morphism
$\texttt{\large{Z}}(\operatorname{Eq}(f,g))\to\texttt{\large{Z}}(B)$ and,
thus, we can consider the unique canonical graph morphism
$\psi:\texttt{\large{Z}}(\operatorname{Eq}(f,g))\to\operatorname{Eq}(\texttt{\large{Z}}(f),\texttt{\large{Z}}(g))$
given by the universal property of the equalizer in Graph.

######  Proposition 3.15.

Let $A,B$ rings and $f,g:A\to B$ ring homomorphisms; then
$\psi:\texttt{\large{Z}}(\operatorname{Eq}(f,g))\to\operatorname{Eq}(\texttt{\large{Z}}(f),\texttt{\large{Z}}(g))$
is a strong graph morphism. Moreover,

  1. 1.

if $\\{\mathcal{R}_{A}\\}$ satisfies the following property: for all $A,B$ in
CRing and $f,g:A\to B$,

$\big{(}a\mathcal{R}_{A}a^{\prime}\Longrightarrow
a\mathcal{R}_{\operatorname{Eq}(f,g)}a^{\prime}\big{)}\text{\, for
all\,}a,a^{\prime}\in\operatorname{Eq}(f,g),$

then $\psi$ is a graph monomorphism;

  2. 2.

if $\\{\mathcal{R}_{A}\\}$ also satisfies the following property: for all
$A,B$ in CRing and $f,g:A\to B$

$\big{(}f(a)\mathcal{R}_{B}\;g(a)\Longrightarrow\exists\;a^{\prime}\in\operatorname{Eq}(f,g)\text{
such that }a\mathcal{R}_{A}a^{\prime}\big{)}\text{\, for all\,}a\in A,,$

then $\psi$ is a graph epimorphism.

###### Proof.

By construction $\psi$ is necessarily defined as
$\psi([a]_{\operatorname{Eq}(f,g)})=[a]_{A}$, and it is immediately seen that
$\psi$ is a strong graph morphism. The hypotheses in 1, resp. 2, are exactly
what is needed for $\psi$ to be injective, resp. surjective. ∎

An example of a zero-divisor functorial family that satifies such properties
is $\\{=_{A}\\}$, but not every zero-divisor functorial family is such; for
instance, the family $\\{\sim_{A}\\}$ does not preserves equalizers, as
observed in [DJS21], Example 3.5. We see next that $\\{=_{A}\\}$ is indeed the
only zero-divisor functorial family that preserves them.

######  Proposition 3.16.

Let $\\{\mathcal{R}_{A}\\}$ be a zero-divisor functorial family; then its
associated zero-divisor functor Z preserves equalizers if and only if
$\mathcal{R}_{A}$ is $=_{A}$ for all $A\in\textbf{CRing}$.

###### Proof.

For every ring $A$ and elements $a,b\in A$ such that $a\mathcal{R}_{A}b$,
consider the unique ring homomorphisms $f,g\,:\,{\mathbb{Z}}[x]\to A$ such
that $f(x)=a$, $g(x)=b$. Since
$\texttt{\large{Z}}(f)([x])=[f(x)]=[a]=[b]=[g(x)]=\texttt{\large{Z}}(g)([x])$,
the equivalence class of the polynomial $x\in{\mathbb{Z}}[x]$ belongs to
$\operatorname{Eq}(\texttt{\large{Z}}(f),\texttt{\large{Z}}(g))$. By
Proposition 3.15 there exists $p(x)\in\operatorname{Eq}(f,g)$ such that
$p(x)\mathcal{R}_{{\mathbb{Z}}[x]}x$. Since $\\{\mathcal{R}_{A}\\}$ is non
degenerate functorial, from Proposition 3.6 we have $(p(x))=(x)$ in
${\mathbb{Z}}[x]$, i.e. $p(x)=\pm x$. Thus $\pm x\in\operatorname{Eq}(f,g)$;
hence $a=\pm b$.

Now, consider the projections maps $\pi_{1},\pi_{2}\,:\,A\times A\to A$; then,
$a\mathcal{R}_{A}b$ implies that $\pi_{1}(a,b)\mathcal{R}_{A}\pi_{2}(a,b)$.
Since Z preserves all equalizers, there exists
$(c,c)\in\operatorname{Eq}(\pi_{1},\pi_{2})$ such that
$(c,c)\mathcal{R}_{A\times A}(a,b)$; hence $(c,c)=\pm(a,b)$, and we may
conclude that $a=b$ and that $\mathcal{R}_{A}$ is $=_{A}$, for all $A$. ∎

As a direct consequence, we have the following.

######  Proposition 3.17.

The only zero-divisor functorial family for which its associated zero-divisor
functor Z preserves all finite limits is the trivial functorial family.

Thus, if $\\{\mathcal{R}_{A}\\}$ is not the trivial functorial family, Z does
not have a left adjoint functor.

Finally, notice that Z never preserves coproducts, independently of the choice
of the family $\\{\mathcal{R}_{A}\\}$, since a zero-divisor graphs is always
connected, $\texttt{\large{Z}}(A\otimes_{\mathbb{Z}}B)$ cannot be written as
coproduct of graphs. Therefore Z does not preserve all finite colimits and it
does not admit a right adjoint functor. Observe that, for every
$A,B\in\textbf{CRing}$ we have a canonical graph morphism
$\texttt{\large{Z}}(A)\sqcup\texttt{\large{Z}}(B)\to\texttt{\large{Z}}(A\otimes_{\mathbb{Z}}B)$,
which is not injective, because $[0]_{A}$ and $[0]_{B}$ both map to
$[0]_{A\otimes B}$, and it is easy to see that in general it is neither
surjective.

###  3.4. An Inversion of Product Theorem

Given a graph $G$ and $v\in V(G)$, one can consider the subgraph of $G$
induced by $N(v)$, where $N(v)\subseteq V(G)$ is the set of adjacents of $v$.
We start with the following easy lemma.

######  Lemma 3.18.

Let $A_{1},A_{2}$ be rings and $G_{1},G_{2}$ the subgraphs of
$\texttt{\large{Z}}(A_{1})\times\texttt{\large{Z}}(A_{2})$ induced by
$N\big{(}[0],[1]\big{)}$ and $N\big{(}[1],[0]\big{)}$, respectively; then,
$G_{1}\simeq\texttt{\large{Z}}(A_{1})$ and
$G_{2}\simeq\texttt{\large{Z}}(A_{2})$.

###### Proof.

Let us show for instance that $G_{1}\simeq\texttt{\large{Z}}(A_{1})$; the
other case is similar. Recall that a vertex $\big{(}[a],[b]\big{)}$ belonging
$\texttt{\large{Z}}(A)\times\texttt{\large{Z}}(B)$ is adjacent to
$\big{(}[0],[1]\big{)}$ if and only if $b=0$. The claim follows directly from
the fact that in a product of graphs adjacency is defined component by
component. ∎

######  Definition 3.19.

We call two elements $a,b\in A$ _orthogonal_ , and we write $a\perp b$, if
$ab=0$ and $\operatorname{Ann}(a)\cap\operatorname{Ann}(b)=0$.

In relation to the study of graphs, orthogonality has been introduced in
[ALS03]; we provided in the above a modification of the original definition,
that will allow us to work with graphs with loops as well. It is easy to see
that two elements $a,b\in A$ are orthogonal if and only if $ab=0$ and $a+b$ is
an $A$-regular, cf. [ALS03], Lemma 3.3.

We are now ready to prove one of our main results.

######  Theorem 3.20.

Let $\\{\mathcal{R}_{A}\\}$ be a zero-divisor functorial family. Let also
$A,B_{1},B_{2}$ be rings such that
$\texttt{\large{Z}}(A)\simeq\texttt{\large{Z}}(B_{1})\times\texttt{\large{Z}}(B_{2})$,
with
$\texttt{\large{Z}}(B_{1}),\texttt{\large{Z}}(B_{2})\not\simeq\texttt{\large{Z}}(\textbf{0})$;
then,

1\. there exist $a_{1},a_{2}\in A$ such that

  1. i)

$a_{1}\perp a_{2}$ and $a_{1},a_{2}\in\mathcal{D}^{*}(A)$;

  2. ii)

$\texttt{\large{Z}}(B_{1})\simeq\texttt{\large{Z}}(\operatorname{Ann}(a_{2}))$
and
$\texttt{\large{Z}}(B_{2})\simeq\texttt{\large{Z}}(\operatorname{Ann}(a_{1}))$;

  3. iii)

there exist strong morphisms
$\psi_{i}\,:\,\texttt{\large{Z}}(B_{i})\longrightarrow\texttt{\large{Z}}(A/\operatorname{Ann}(a_{i}))$,
with $i=1,2$.

2\. Let $i,j\in\\{1,2\\}$ with $i\neq j$. If
$\bar{x}\mathcal{R}_{A/\operatorname{Ann}(a_{i})}\bar{x}^{\prime}$ implies
that $x\mathcal{R}_{A}x^{\prime}$ for all
$x,x^{\prime}\in\operatorname{Ann}(a_{j})$, then $\psi_{i}$ is injective;

3\. if $\operatorname{Ann}(a_{1})+\operatorname{Ann}(a_{2})=(1)$, then
$A\simeq A/\operatorname{Ann}(a_{1})\times A/\operatorname{Ann}(a_{2})$ and
$\psi_{1},\psi_{2}$ are surjective.

###### Proof.

1\. Let
$\varphi\,:\,\texttt{\large{Z}}(A)\longrightarrow\texttt{\large{Z}}(B_{1})\times\texttt{\large{Z}}(B_{2})$
a graph isomorphism and let us consider $a_{1},a_{2}\in A$ such that
$[a_{1}]=\varphi^{-1}([1],[0])$ and $[a_{2}]=\varphi^{-1}([0],[1])$. Since
$([1],[0]),\,([0],[1])\in
V(\texttt{\large{Z}}(B_{1})\times\texttt{\large{Z}}(B_{2}))$ are adjacent and
$\varphi$ is a strong morphism, we have that $[a_{1}]=\varphi^{-1}([1],[0])$
and $[a_{2}]=\varphi^{-1}([0],[1])$ are adjacent as well, i.e. $a_{1}a_{2}=0$.

Since $[0]$ is the only vertex adjacent to all the others, one has necessarily
$\varphi([0])=([0],[0])$. Thus, if $a_{1}$ were zero, we would have that
$\varphi^{-1}([1],[0])=[a_{1}]=\varphi^{-1}([0],[0])$ which implies
$B_{1}\simeq\textbf{0}$ and, in turn,
$\texttt{\large{Z}}(B_{1})\simeq\texttt{\large{Z}}(\textbf{0}),$ a
contradiction. The case $a_{2}=0$ is analogous and we have proven the first
part of the claim.

Now, let $x\in\operatorname{Ann}(a_{1})\cap\operatorname{Ann}(a_{2})$ and
$b_{1}\in B_{1}$, $b_{2}\in B_{2}$ such that $\varphi([x])=([b_{1}],[b_{2}])$.
Since $[x]$ is adjacent to both $[a_{1}]$ and $[a_{2}]$, we have that
$\varphi([x])$ is adjacent to both $\varphi([a_{1}])=([1],[0])$ and
$\varphi([a_{2}])=([0],[1])$. Thus, from the very definition of adjacency of
the product it follows that $b_{1}=0=b_{2}$, which implies that
$[x]=\varphi^{-1}([0],[0])=[0]$ and $x=0$, and we proved i).

By our choice of $a_{1},a_{2}$ and the fact that $\varphi$ is an isomorphism,
the conclusion of ii) is a straightforward application of Lemma 3.18.

iii) By symmetry, we may prove the assertion for $i=1$. By what was proved in
ii), we only need to show that there exists a morphism
$\psi_{1}^{\prime}\,:\,\texttt{\large{Z}}(\operatorname{Ann}(a_{2}))\longrightarrow\texttt{\large{Z}}(A/\operatorname{Ann}(a_{1}))$
which is strong. Let us then define $\psi_{1}^{\prime}\big{(}[x]\big{)}$ as
$[\bar{x}]$. The map $\psi_{1}^{\prime}$ results to be a strong morphism,
since $\operatorname{Ann}(a_{1})\cap\operatorname{Ann}(a_{2})=0$: taken
$x_{1},x_{2}\in\operatorname{Ann}(a_{2})$, one has that $x_{1}x_{2}=0$ if and
only if $x_{1}x_{2}\in\operatorname{Ann}(a_{1})$.

2\. By symmetry, we may prove the assertion for $i=1$. From what was proved in
1.ii), we only need to show that $\psi_{1}^{\prime}$ is injective, and that is
straightforward.

3\. By the Chinese Remainder Theorem one has $A\simeq A/(0)\simeq
A/\operatorname{Ann}(a_{1})\times A/\operatorname{Ann}(a_{2})$. Observe that
the maps $\psi_{i}^{\prime}$ defined in the proof of 1.iii) are also
surjective. ∎

######  Remark 3.21.

The above theorem has been inspired by and extends [DJS21], Theorem 3.6. Its
proof contains a mistake, since $\Theta$ is applied to an ideal of a
commutative ring.

  1. 1.

Observe that, in Theorem 3.20.1.ii), $\operatorname{Ann}(a_{2})$ is determined
once $a_{1}$ is given, since if $a,b,c\in A$ are such that $a\perp b$ and
$a\perp c$ then $b\asymp c$, which is easily seen.

  2. 2.

When $\mathcal{R}_{A}=\sim_{A}$, then the hypothesis of Part 2. holds true.
Let $x_{1},x_{2}\in\operatorname{Ann}(a_{2})$ be such that
$\bar{x}_{1}\mathcal{R}_{A/\operatorname{Ann}(a_{i})}\bar{x}_{2}$. Since
$\sim$ is the coarsest of such relations, see Remark 3.9, there exist $h,k\in
A$ such that $x_{1}-hx_{2}=0=x_{2}-kx_{1}$ in $A/\operatorname{Ann}(a_{1})$
and, therefore, $x_{1}-hx_{2},x_{2}-kx_{1}\in\operatorname{Ann}(a_{1})$. Since
$x_{1},x_{2}\in\operatorname{Ann}(a_{2})$, one has
$x_{1}-hx_{2}=0=x_{2}-kx_{1}$ in $A$, i.e. $(x_{1})=(x_{2})$.

Recall that, given a ring $A$, the set $S=A\smallsetminus\mathcal{D}(A)$ is
multiplicatively closed; $A$ is said to be total if $A$ is isomorphic to its
total ring of fractions $S^{-1}A$; this is equivalent to require that every
regular element of $A$ is invertible.

######  Corollary 3.22.

Let $A$ be total, $\mathcal{R}_{A}=\sim_{A}$ and $B_{1},B_{2}$ rings such that
$\texttt{\large{Z}}(A)\simeq\texttt{\large{Z}}(B_{1})\times\texttt{\large{Z}}(B_{2})$,
with
$\texttt{\large{Z}}(B_{1}),\texttt{\large{Z}}(B_{2})\not\simeq\texttt{\large{Z}}(\textbf{0})$;
then $A\simeq A_{1}\times A_{2}$, where $A_{1}$, $A_{2}$ are quotients of $A$
such that $\texttt{\large{Z}}(A_{1})\simeq\texttt{\large{Z}}(B_{1})$ and
$\texttt{\large{Z}}(A_{2})\simeq\texttt{\large{Z}}(B_{2})$.

###### Proof.

By Theorem 3.20.1 it follows that there exist
$a_{1},a_{2}\in\mathcal{D}^{*}(A)$ such that $a_{1}+a_{2}$ is a regular
element and, since $A$ is total, there exists $c\in A$ such that
$c(a_{1}+a_{2})=1$. Moreover, $a_{1}\in\operatorname{Ann}(a_{2})$ and
$a_{2}\in\operatorname{Ann}(a_{1})$, that implies that
$\operatorname{Ann}(a_{1})+\operatorname{Ann}(a_{2})=(1)$; the conclusion
follows now by Remark 3.21.2 and Theorem 3.20.2 and 3. ∎

######  Example 3.23.

Consider $=\mathbb{F}_{2}[x,y]/(xy)$; then
$S=A\smallsetminus(\bar{x},\bar{y})$ and $S^{-1}A$ is not a total ring. It
holds that
$\texttt{\large{Z}}(S^{-1}A,\sim)\simeq\texttt{\large{Z}}(\mathbb{F}_{2}[x],\sim)\times\texttt{\large{Z}}(\mathbb{F}_{2}[y],\sim)\simeq\texttt{\large{Z}}({\mathbb{Z}},\sim)\times\texttt{\large{Z}}({\mathbb{Z}},\sim)$,
but $S^{-1}A$ is local and, thus, does not admit a non trivial factorization
as a direct product.

We conclude this section with a result about Z and localization.

######  Proposition 3.24.

Let $A$ be a ring, $S\subset A$ multiplicatively closed and
$\varphi_{S}\,:\,A\longrightarrow S^{-1}A$ be the canonical homomorphism; then

  1. 1.

if $\frac{a}{s}\mathcal{R}\frac{a}{1}$ for all $a\in A$, $s\in S$, then
$\texttt{\large{Z}}(\varphi_{S})$ is a graph epimorphism;

  2. 2.

if $S$ consists of regular elements, then $\texttt{\large{Z}}(\varphi_{S})$ is
a graph comorphism.

###### Proof.

1\. Since $\frac{a}{s}\mathcal{R}\frac{a}{1}$ for all $a\in A$ and $s\in S$,
every element in $V(\texttt{\large{Z}}(S^{-1}(A))$ is the image by
$\texttt{\large{Z}}(\varphi_{S})$ of a vertex in $\texttt{\large{Z}}(A)$, as
desired.

2\. Let now $S\subseteq A\smallsetminus\mathcal{D}(A)$, and $a,b\in A$ such
that $\frac{a}{1}\frac{b}{1}=0$; then, there exists a regular element $c$ such
that $cab=0$. Therefore $ab=0$ and $\texttt{\large{Z}}(\varphi_{S})$ is a
comorphism. ∎

Note that the trivial family of relations does not satisfy $1.$ of the above
proposition.

######  Corollary 3.25.

Let $\\{\mathcal{R}_{A}\\}=\\{\sim_{A}\\}$; then, for all total rings
$A\in\textbf{CRing}$ and multiplicatively closed sets $S\subseteq A$, we have
that $\texttt{\large{Z}}(\varphi_{S})$ is a strong graph epimorphism. In
particular, $\texttt{\large{Z}}(S^{-1}A)$ as a strong quotient of
$\texttt{\large{Z}}(A)$.

Observe that, in general, $\texttt{\large{Z}}(\varphi_{S})$ is not an
isomorphism, even when $\varphi_{S}$ is injective.

##  4\. Applications to the finite case

It would be desirable to have a structure theorem for
$\texttt{\large{Z}}(A)=\texttt{\large{Z}}(A,\mathcal{R}_{A})$ in case this is
finite. Observe that its finitess depends on the chosen relation, as for
instance $\texttt{\large{Z}}({\mathbb{Q}},=)$ is infinite, whereas
$\texttt{\large{Z}}({\mathbb{Q}},\sim)$ is finite. Clearly, when $A$ is finite
then $\texttt{\large{Z}}(A)$ is finite, indipendently of the given
$\mathcal{R}_{A}$; moreover, we know by Proposition 3.6 and Remark 2.5 then
when $\texttt{\large{Z}}(A)$ is finite then $\texttt{\large{Z}}(A,\sim)$ is
finite. Therefore, if we give, and we do, a characterization in the case
$\mathcal{R}_{A}=\sim_{A}$, then we will have the largest family of rings for
which $\texttt{\large{Z}}(A)$ might be finite.

For the rest of the section let us fix $\\{\mathcal{R}_{A}\\}=\\{\sim_{A}\\}$.
First of all, notice that, by definition, $\texttt{\large{Z}}(A)$ is finite
exactly when $A$ has finitely many principal ideals, which immediately implies
that $A$ is artinian.

######  Theorem 4.1.

Let $A$ be a ring. Then, the following are equivalent:

  1. 1.

$\texttt{\large{Z}}(A)$ is finite;

  2. 2.

$A$ has finitely many ideals;

  3. 3.

$A\simeq A_{1}\times A_{2}$, where $A_{1}$ is an artinian PIR and $A_{2}$ is a
finite ring.

###### Proof.

$1\Rightarrow 2$: We already observed that $\mathcal{PI}_{A}$ is finite and
$A$ artinian; since every element of $\mathcal{I}_{A}$ is a finite sum of
principal ideals, the conclusion follows.

$2\Rightarrow 3$: This is contained in [Z14], Theorem 3.3. Observe that $A$ is
artinian and can be written as a finite product of artinian local rings, which
all have a finite number of ideals. It is thus sufficient to show that each
local factor of $A$ is either a PIR or a finite ring. By contradiction, say
that a factor $(B,\mathfrak{m},k)$ of $A$ is infinite and not PIR. It is well
known that a local ring is finite if and only if $A$ is artinian with finite
residue field and this is principal if and only if the $k$-vector space
$\mathfrak{m}/\mathfrak{m}^{2}$ has dimension $>1$. We may thus conclude that
$Q$ has infinitely many ideals, which is the sought after contradicion.

$3\Rightarrow 1$: By Corollary 3.12 we have that
$\texttt{\large{Z}}(A)\simeq\texttt{\large{Z}}(A_{1})\times\texttt{\large{Z}}(A_{2})$,
where $\texttt{\large{Z}}(A_{2})$ is finite. We only need to show that
$\texttt{\large{Z}}(A_{1})$ is finite as well. Since an artinian PIR is a
finite product of local artinian PIRs, we may assume that $A_{1}$ is local.
Thus, every ideal of $A_{1}$ can be written as $(p^{i})$, for some $p\in
A_{1}$, and $p^{n}=0$ for some natural number $n\geqslant 1$ from which it
follows that $\mathcal{PI}_{A}$ is finite. ∎

The previous theorem, together with Corollary 3.12, yields a reduction to the
case of local artinian rings, since we can reconstruct $\texttt{\large{Z}}(A)$
from the graphs which are associated with its local factors.

A family of graphs which turns out to be important to this end is staircases
graphs, i.e. those whose adjacency matrix is an ascending staircase of 1’s. In
other words, a graph with vertex set $V=\\{0,\dots,k\\}$ is called staircase
graph of index $k$, and it is denoted by $SG_{k}$, when

$i\text{\, is adjacent to\, }j\text{\,\, if and only if \,\,}i+j\geqslant k.$

Observe that in $SG_{k}$ it holds that $\\{k\\}=N(0)\subset
N(1)\subset\dots\subset N(k)=\\{0,\dots,k\\}$ and, thus,
$1=\deg(0)<\deg(1)<\dots<\deg(k)=k+1$.

These graphs are uniquely determined, up to isomorphisms, by the degree of the
vertices. In fact, in [DJS21] Proposition 4.1 the following was proved: a
graph $G=(\\{v_{0},\ldots,v_{k}\\},E)$, with $k>0$, such that
$\deg(v_{i})\in\\{1,\ldots,k+1\\}$ for each $i$ and
$\deg(v_{i})\neq\deg(v_{j})$ whenever $i\neq j$, is isomorphic to $SG_{k}$.
Moreover, since $SG_{k}\simeq\texttt{\large{Z}}({\mathbb{Z}}/(p^{k}))$ and
${\mathbb{Z}}/(p^{k})$ is a local ring, any $A$ such that
$\texttt{\large{Z}}(A)\simeq SG_{k}$ is local as well, cf. [DJS21],
Proposition 4.2 and Corollary 5.1.

Before we proceed with the next lemma, we observe the following. Fix $x\in A$;
the map $m_{x}:\texttt{\large{Z}}(A)\rightarrow\texttt{\large{Z}}(A)$ defined
as $(a)\mapsto(xa)$ is a graph morphism. Moreover, for every $z\in A$ we have
that $(z)\in N((xa))\iff z\in\operatorname{Ann}(xa)\iff
zx\in\operatorname{Ann}(a)\iff(zx)\in N((a))$. Thus, if $a,b\in A$ are such
that $N((a))\subseteq N((b))$ then $N((xa))\subseteq N((xb)).$

######  Lemma 4.2.

Let $A$ be a ring, $a\in A$ and
$\varphi\,:\,SG_{k}\longrightarrow\texttt{\large{Z}}(A)$ a surjective strong
morphism, for some integer $k\geqslant 0$; then, for all $0\leqslant
i\leqslant j\leqslant k$, it holds that $N(x\varphi(i))\subseteq
N(x\varphi(j))$.

###### Proof.

It is enough to prove that $N(\varphi(i))\subseteq N(\varphi(j))$. By
hypothesis, every vertex in $N(\varphi(i))$ is the image of a vertex
$l\in\\{0,\dots,k\\}$; since $\varphi$ is a comorphism, $l$ is adjacent to $i$
and $i\leqslant j$ implies that $l\in N(i)\subseteq N(j)$. Thus,
$\varphi(l)\in N(\varphi(j))$. ∎

######  Lemma 4.3.

Let $A\neq\textbf{0}$ be a ring such that $\mathcal{D}(A)=\mathcal{N}(A)$;
then, for all $a,b\in\mathcal{D}^{*}(A)$ then
$\operatorname{Ann}(a)\cup\operatorname{Ann}(b)\subsetneq\operatorname{Ann}(ab)$.

###### Proof.

It is well-known that
$\operatorname{Ann}(a)\cup\operatorname{Ann}(b)\subseteq\operatorname{Ann}(ab)$;
therefore, let us assume by contradiction that there exist
$a,b\in\mathcal{D}^{*}(A)$ such that
$\operatorname{Ann}(a)\cup\operatorname{Ann}(b)=\operatorname{Ann}(ab)$ and
let $n,m\geqslant 1$ be the nilpotency indexes of $a$ and $b$, respectively.
By inducting on $k+l$, one can easily show first that $a^{k}b^{l}\neq 0$ for
all $k\in\\{0,\dots,n-1\\}$, $l\in\\{0,\dots,m-1\\}$, and secondly that
$\operatorname{Ann}(a^{k}b^{l})=\operatorname{Ann}(a^{k})\cup\operatorname{Ann}(b^{l})$,
for all such $k,l$.

Now, if we let $\alpha=a^{n-1}$ and $\beta=b^{m-1}$, then
$\alpha,\,\beta,\,\alpha\beta\neq 0$ and $\alpha^{2}=\beta^{2}=0$;
accordingly,
$(\alpha+\beta)\in\operatorname{Ann}(\alpha\beta)=\operatorname{Ann}(\alpha)\cup\operatorname{Ann}(\beta)$
and, thus, $\alpha\beta=0$, contradiction. ∎

As an easy consequence, one characterizes local artinian rings by means
annihilator ideals, cf. [DJS21], Theorem 5.3.

######  Corollary 4.4.

Let $A\neq\textbf{0}$ be artinian; then $A$ is local if and only if for all
$a,b\in\mathcal{D}^{*}(A)$ it holds that
$\operatorname{Ann}(a)\cup\operatorname{Ann}(b)\subsetneq\operatorname{Ann}(ab)$.

###### Proof.

If $A$ were not local, the Structure Theorem for artinian rings would yields
that $A\simeq A_{1}\times A_{2}$, for some rings $A_{1},A_{2}\neq\textbf{0}$;
also, the elements $a=b=(1,0)$ would contradict the hypothesis. Viceversa,
when $A$ is local artinian one has
$\mathcal{D}(A)=\mathcal{N}(A)=\mathfrak{m}$, and the conclusion follows
directly by the previous lemma. ∎

Recall that, when $A$ is local, one has that $(a),(b)\in\mathcal{PI}_{A}$ are
equal if and only if $a\approx b$.

######  Lemma 4.5.

Let $A$ be a local artinian ring and
$\varphi\,:\,SG_{k}\longrightarrow\texttt{\large{Z}}(A)$ be an isomorphism for
some $k>0$.

  1. 1.

Let $x\in A$; if there exist $0\leqslant i<j\leqslant k$ such that
$\operatorname{Ann}(x\varphi(i))=\operatorname{Ann}(x\varphi(j))$, then
$x\varphi(i)=x\varphi(j)=0;$

  2. 2.

$\varphi(i)=\varphi(1)^{i}$, for all $i\in\\{0,\dots,k\\}.$

###### Proof.

For all $i,j\in\\{0,\dots,k\\}$, if $\varphi(i)=\varphi(j)$ then
$N(\varphi(i))=N(\varphi(j))$, i.e.
$\operatorname{Ann}(\varphi(i))=\operatorname{Ann}(\varphi(j))$, and the
viceversa holds since $\varphi$ is an isomorphism. The graph $SG_{k}$ is
staircase and, thus, $i<j$ if and only if $N(i)\subset N(j)$; this is
equivalent to $N(\varphi(i))\subset N(\varphi(j))$, which in turn means
$\operatorname{Ann}(\varphi(i))\subset\operatorname{Ann}(\varphi(j))$. For all
$i\in\\{0,\dots,k\\}$, let $\varphi(i)=(a_{i}).$

1\. Assume that $\operatorname{Ann}(xa_{i})=\operatorname{Ann}(xa_{j})$; then
$(xa_{i})=(xa_{j})$ and there exists $u\in A^{*}$ such that
$a_{i}-ua_{j}\in\operatorname{Ann}(x)$. Moreover, since $i<j$, we have that
$\operatorname{Ann}(a_{i})\subset\operatorname{Ann}(a_{j})$ and there exists
$y\in\operatorname{Ann}(a_{j})\smallsetminus\operatorname{Ann}(a_{i})$; thus,
$(a_{i}-ua_{j})y=a_{i}y\neq 0$ and it follows that
$a_{i}-ua_{j}\not\in\operatorname{Ann}(y)$.

Since $\varphi^{-1}\big{(}(x)\big{)},\varphi^{-1}\big{(}(y)\big{)}\in
V(SG_{k})$, we have that either $N(\varphi^{-1}\big{(}(x)\big{)})\subseteq
N(\varphi^{-1}\big{(}(y)\big{)})$, or
$N(\varphi^{-1}\big{(}(x)\big{)})\supseteq N(\varphi^{-1}\big{(}(y)\big{)})$
and, accordingly, either $\operatorname{Ann}(x)\subseteq\operatorname{Ann}(y)$
or $\operatorname{Ann}(y)$ $\subseteq\operatorname{Ann}(x)$. Since
$a_{i}-ua_{j}\in\operatorname{Ann}(x)\smallsetminus\operatorname{Ann}(y)$, we
must have $\operatorname{Ann}(y)\subset\operatorname{Ann}(x)$ and, by our
choice of $y$, it follows that $0=x(a_{j})=x(a_{i})$.

2\. The cases $i=0,1$ are easy: $(a_{1})\\!=\\!(a_{1})^{1}$ and, since
$\operatorname{Ann}(a_{0})\\!=0\\!=\\!\operatorname{Ann}(1)$, it follows that
$(a_{0})\\!=\\!(1)\\!=\\!(a_{1})^{0}$. Thus, if $k=1$ we are done and may
assume $k>1$. We are going to prove that $(a_{i+1})=(a_{1})(a_{i})$ for all
$1\leqslant i<k$, and that implies the conclusion.

First of all, observe that $N(j)\\!\neq\\!N(0)$ for all
$j\\!\in\\!\\{1,\dots,k\\!-\\!1\\}$. This implies that
$\operatorname{Ann}(a_{j})\neq\operatorname{Ann}(a_{0})=0$ and, thus,
$a_{j}\in\mathcal{D}(A)$; moreover, $(a_{j})\\!\neq\\!(a_{k})=0$, i.e.
$a_{j}\in\mathcal{D}^{*}(A)$. By Corollary 4.4, we have that
$\operatorname{Ann}(a_{1}a_{j})\\!\supset\\!\operatorname{Ann}(a_{j})$ and
$N((a_{1}a_{j}))\\!\supset\\!N((a_{j}))$; therefore
$\deg((a_{1}a_{j}))\\!>\\!\deg((a_{j}))\\!=\\!j+1$ for all
$j\\!\in\\!\\{1,\dots,k\\!-\\!1\\}$.

Now take $j,l\in\\{0,\dots,k\\}$ with $j\neq l$; by 1, if
$\operatorname{Ann}(a_{1}a_{j})=\operatorname{Ann}(a_{1}a_{l})$ then
$a_{1}a_{j}=a_{1}a_{l}=0$ and, since $\varphi$ is an isomorphism, the only
possible cases are when $j,l\in\\{k\\!-\\!1,k\\}$. Moreover, by Lemma 4.2 we
get that
$N((a_{1}a_{i}))\\!\subset\\!N((a_{1}a_{i+1}))\\!\subset\\!N((a_{1}a_{i+2}))\\!\subset\dots\subset\\!N((a_{1}a_{k-2}))\\!\subset\\!N((a_{1}a_{k-1}))=N((a_{1}a_{k}))=\mathcal{PI}_{A}$,
where the last equality is due to the fact that $a_{k}=0$. Taking the degrees
in the above chain of inclusions, we obtain
$\deg((a_{1}a_{i}))\,<\,\deg((a_{1}a_{i+1}))\,<\,\deg((a_{1}a_{i+2}))\,<\,\dots\,<\,\deg((a_{1}a_{k-2}))\,<\,\deg((a_{1}a_{k-1}))=\deg((a_{1}a_{k}))=k+1$.
We know that $\deg((a_{1}a_{i}))\\!>\\!i+1$; on the other hand
$\deg((a_{1}a_{i}))\leqslant i+2$, otherwise we would have
$\deg((a_{1}a_{k-1}))\\!>\\!i+2+(k\\!-\\!1-i)=k+1$, which is a contradiction.
Therefore, $(a_{1}a_{i})$ has degree $i+2$ and, therefore,
$(a_{1}a_{i})=(a_{i+1})$, as desired.

∎

######  Proposition 4.6.

Let $(A,\mathfrak{m})$ be a local artinian ring. Then, $A$ is PIR if and only
if $\texttt{\large{Z}}(A)\simeq SG_{k}$, where $k$ is the nilpotency index of
$\mathfrak{m}$.

###### Proof.

Since $A$ is PIR, it is well-known that every ideal $I\subseteq A$ can be
written as $I=\mathfrak{m}^{i}$ and $\mathfrak{m}^{i}$ is adjacent to
$\mathfrak{m}^{j}$ if and only if $i+j\geqslant k$, where $k$ is the
nilpotency index of $\mathfrak{m}$; it is easy to conclude that the map
$\varphi\,:\,SG_{k}\longrightarrow\texttt{\large{Z}}(A)$ defined by
$\varphi(i)=\mathfrak{m}^{i}$ is an isomorphism.

Viceversa, by Lemma 4.5.2, every principal ideal is a power of a fixed
principal ideal $(x)$; since $A$ is noetherian, it follows that every ideal is
a power of $(x)$ and that $A$ is PIR. ∎

######  Example 4.7.

Let $A={\mathbb{Z}}/(4)[x]/(2x,x^{2}-2)$ and
$B=\mathbb{F}_{2}[x,y]/(x^{2},xy,y^{2})$. Then, $A$ is a local ring which is
PIR by Proposition 4.6, since $\texttt{\large{Z}}(A)\simeq SG_{3}$; since
$\texttt{\large{Z}}(\mathbb{F}_{2}[x,y]/(x^{2},xy,y^{2}))$ has four distinct
vertices with a loop, namely $(\bar{x})$, $(\bar{y})$, $(\bar{x}+\bar{y})$ and
$(\bar{0})$, in contrast to $SG_{3}$ which has only two vertices with a loop.

######  Theorem 4.8.

An artinian ring $A$ is PIR if and only if $\texttt{\large{Z}}(A)$ is
isomorphic to a finite product of staircases graphs.

###### Proof.

If $A$ is PIR, then the conclusion follows by a straightforward application of
Proposition 4.6 and Corollary 3.12.

Viceversa, if
$\texttt{\large{Z}}(A)\simeq\mathop{\mathchoice{\hbox{\zzzsyb\char
2\relax}}{\hbox{\zzzsy\char
2\relax}}{\scriptstyle\times}{\scriptscriptstyle\times}}\limits_{i=1}^{n}SG_{k_{i}}$,
then $\texttt{\large{Z}}(A)\simeq\mathop{\mathchoice{\hbox{\zzzsyb\char
2\relax}}{\hbox{\zzzsy\char
2\relax}}{\scriptstyle\times}{\scriptscriptstyle\times}}\limits_{i=1}^{n}\texttt{\large{Z}}({\mathbb{Z}}/(p_{i})^{k_{i}})$.
Since an artinian ring is total, Corollary 3.22 implies that
$A\simeq\mathop{\mathchoice{\hbox{\zzzsyb\char 2\relax}}{\hbox{\zzzsy\char
2\relax}}{\scriptstyle\times}{\scriptscriptstyle\times}}\limits_{i=1}^{n}A_{i}$
where $\texttt{\large{Z}}(A_{i})\simeq SG_{k_{i}}$. Each $A_{i}$ is
necessarily artinian; if $A_{i}$ were not local for a certain $i$, we would
have $A_{i}\simeq B_{1}\times B_{2}$ with $B_{1},B_{2}\not\simeq\textbf{0}$,
$\texttt{\large{Z}}({\mathbb{Z}}/(p_{i})^{k_{i}})\simeq\texttt{\large{Z}}(A_{i})\simeq\texttt{\large{Z}}(B_{1})\times\texttt{\large{Z}}(B_{2})$
by Corollary 3.12, and another application of Corollary 3.22 would imply that
also ${\mathbb{Z}}/(p_{i})^{k_{i}}$ can be factorized, contradiction. Finally,
each of the $A_{i}$ is PIR, as yielded by Proposition 4.6. ∎

######  Remark 4.9.

Keeping in mind Theorem 4.1, in order to complete the picture, one should
understand the structure of $\texttt{\large{Z}}(A)$ when $A$ is a finite ring,
which is not PIR. This does not seem to be an easy task, since if $A$ and $B$
are rings such that the posets $\mathcal{I}_{A}$ and $\mathcal{I}_{B}$ are
isomorphic, it is not true in general that
$\texttt{\large{Z}}(A)\simeq\texttt{\large{Z}}(B)$. For instance consider
$A=\mathbb{F}_{2}[x,y]/(x^{2},y^{2})$ and
$B=\mathbb{F}_{2}[x,y]/(xy,x^{2}-y^{2})$; then
$\mathcal{I}_{A}\simeq\mathcal{I}_{B}$ as posets but every vertex in
$\texttt{\large{Z}}(A)$ has a loop whereas $(\bar{x})\in\mathcal{PI}_{B}$ has
none; therefore $\texttt{\large{Z}}(A)\not\simeq\texttt{\large{Z}}(B)$.

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