Narrow Results

The notion of the metric dimension of a graph is well-known and its study is well entrenched in the literature. In this paper, we introduce new classes of graphs that exhibit remarkable characteristic: the metric dimension and the diameter of the unit graph are equal. Additionally, we provide a characterization of finite commutative rings $R$, wherein the metric dimension assumes a value $k$, where $1\le k\le 3$. Further, we also demonstrate an exhaustive examination that ascertains the precise finite commutative rings $R$, in which domination number is equal to metric dimension of unit graph.

Let R be a commutative ring and 𝔸(R) be the set of ideals with nonzero annihilators. The annihilating-ideal graph of R is defined as the graph 𝔸𝔾(R) with the vertex set 𝔸(R)* = 𝔸\{(0)} and two distinct vertices I and J are adjacent if and only if IJ = (0). We investigate commutative rings R whose annihilating-ideal graphs have positive genus γ(𝔸𝔾(R)). It is shown that if R is an Artinian ring such that γ(𝔸𝔾(R)) < ∞, then either R has only finitely many ideals or (R, 𝔪) is a Gorenstein ring with maximal ideal 𝔪 and v.dim_R/𝔪𝔪/𝔪² = 2. Also, for any two integers g ≥ 0 and q > 0, there are only finitely many isomorphism classes of Artinian rings R satisfying the conditions: (i) γ(𝔸𝔾(R)) = g and (ii) |R/𝔪| ≤ q for every maximal ideal 𝔪 of R. Also, it is shown that if R is a non-domain Noetherian local ring such that γ(𝔸𝔾(R)) < ∞, then either R is a Gorenstein ring or R is an Artinian ring with only finitely many ideals.

This paper is concerned with investigating the center and radius of zero-divisor and ideal-divisor graphs of finite commutative rings. In particular, we examine the interplay between the center and cut-sets of zero-divisor graphs. Of interest are conditions that need to be met in order for either of these sets to contain the other. We then explore the radius and central sets of ideal-divisor graphs, particularly the classification of the center via radius results.

Let R ⊆ S be a unital extension of commutative rings, with the integral closure of R in S, such that there exists a finite maximal chain of rings from R to S. Then S is a P-extension of R, is a normal pair, each intermediate ring of R ⊆ S has only finitely many prime ideals that lie over any given prime ideal of R, and there are only finitely many -subalgebras of S. Each chain of rings from R to S is finite if dim(R) = 0; or if R is a Noetherian (integral) domain and S is contained in the quotient field of R; or if R is a one-dimensional domain and S is contained in the quotient field of R; but not necessarily if dim(R) = 2 and S is contained in the quotient field of R. Additional domain-theoretic applications are given.

For a prime ideal $P$ of a commutative ring $A$ with identity, we denote (as usual) by $O_P$ its zero-component; that is, the set of members of $P$ that are annihilated by nonmembers of $P$. We study rings in which $O_P$ is an essential ideal, whenever $P$ is an essential prime ideal. We characterize them in terms of the lattices (which are, in fact, complete Heyting algebras) of their radical ideals. We prove that the classical ring of quotients of any ring of this kind is itself of this kind. We show that direct products of rings of this kind are themselves of this kind. We observe that the ring of real-valued continuous functions on a Tychonoff space is of this kind precisely when the underlying set of the space is infinite. Replacing $O_P$ with the pure part of $P$, we obtain a formally stronger variant which is still characterizable in terms of the lattices of radical ideals.

This paper is a sequel to [L. Positselski, Dedualizing complexes and MGM duality, J. Pure Appl. Algebra220(12) (2016) 3866–3909, arXiv:1503.05523 [math.CT]; Contraadjusted modules, contramodules, and reduced cotorsion modules, preprint (2016), arXiv:1605.03934 [math.CT]]. We extend the classical Harrison–Matlis module category equivalences to a triangulated equivalence between the derived categories of the abelian categories of torsion modules and contramodules over a Matlis domain. This generalizes to the case of any commutative ring $R$ with a fixed multiplicative system $S$ such that the $R$-module $S^{-1}R$ has projective dimension $1$. The latter equivalence connects complexes of $R$-modules with $S$-torsion and $S$-contramodule cohomology modules. It takes a nicer form of an equivalence between the derived categories of abelian categories when $S$ consists of nonzero-divisors or the $S$-torsion in $R$ is bounded.

The regular graph of ideals of a commutative ring $R$, denoted by ${\mathrm{\Gamma }}_{\mathrm{\text{reg}}}(R)$, is a graph whose vertex-set is the set of all nontrivial ideals of $R$ and, for every two distinct vertices $I$ and $J$, there is an edge between $I$ and $J$, whenever $I$ contains a nonzero zero-divisor on $J$ or vice versa. In this paper, we will show that ${\mathrm{\Gamma }}_{\mathrm{\text{reg}}}(T(R))$ is isomorphic to an induced sub-graph of ${\mathrm{\Gamma }}_{\mathrm{\text{reg}}}(R)$ which nicely can describe ${\mathrm{\Gamma }}_{\mathrm{\text{reg}}}(R)$. This approach enables us to find the independence number of ${\mathrm{\Gamma }}_{\mathrm{\text{reg}}}(R)$ when $R$ is a reduced ring. Among other things, some basic graph theoretic properties of ${\mathrm{\Gamma }}_{\mathrm{\text{reg}}}(T(R))$ and relevant ring theoretic properties of $R$ will be studied.

Let $R$ be a commutative ring, ${ℐ}(R)$ the set of all ideals of $R$ and $S$, a subset of ${{ℐ}}^{\ast }(R)={ℐ}(R)\backslash \{0\}$. The Cayley sum graph of ideals of $R$, denoted by Cay$({ℐ}(R),S)$, is a simple undirected graph with vertex set is the set ${ℐ}(R)$ and, for any two distinct vertices $I$ and $J$ are adjacent if and only if $I+K=J$ or $J+K=I$, for some $K$ in $S$. In this paper, we study connectedness, Eulerian and Hamiltonian properties of Cay$({ℐ}(R),\mathrm{\text{Max}}(R))$. Moreover, we characterize all commutative Artinian rings $R$ whose Cay$({ℐ}(R),\mathrm{\text{Max}}(R))$ is toroidal.

Let $R$ be a commutative ring with nonzero identity and $J(R)$ be the Jacobson radical of $R$. The Jacobson graph of $R$, denoted by ${𝔍}_R$, is a graph with vertex-set $R\setminus J(R)$, such that two distinct vertices $a$ and $b$ in $R\setminus J(R)$ are adjacent if and only if $1-ab$ is not a unit of $R$. The goal in this paper is to list every finite commutative ring $R$ with nonzero identity (up to isomorphism) such that the graph ${𝔍}_R$ is projective.

Let $R$ be a finite commutative ring with nonzero identity and $U(R)$ be the set of all units of $R.$ The graph $\mathrm{\Gamma }$ is the simple undirected graph with vertex set $R$ in which two distinct vertices $x$ and $y$ are adjacent if and only if there exists a unit element $u$ in $U(R)$ such that $x+uy$ is a unit in $R.$ In this paper, we obtain degree of all vertices in $\mathrm{\Gamma }$ and in turn provide a necessary and sufficient condition for $\mathrm{\Gamma }$ to be Eulerian. Also, we give a necessary and sufficient condition for the complement $\overline{\mathrm{\Gamma }}$ to be Eulerian, Hamiltonian and planar.

In this paper, we initiate the study of the total zero-divisor graph over a commutative ring with unity. This graph is constructed by both relations that arise from the zero-divisor graph and from the total graph of a ring and give a joint insight of the structure of zero-divisors in a ring. We characterize Artinian rings with the connected total zero-divisor graphs and give their diameters. Moreover, we compute major characteristics of the total zero-divisor graph of the ring ${\mathbb{Z}}_m$ of integers modulo $m$ and prove that the total zero-divisor graphs of ${\mathbb{Z}}_m$ and ${\mathbb{Z}}_n$ are isomorphic if and only if $m=n$.

Given any minimal ring extension $k\subset L$ of finite fields, several families of examples are constructed of a finite local (commutative unital) ring $A$ which is not a field, with a (necessarily finite) inert (minimal ring) extension $A\subset B$ (so that $B$ is a separable $A$-algebra), such that $A\subset B$ is not a Galois extension and the residue field of $A$ (respectively, $B$) is $k$ (respectively, $L$). These results refute an assertion of G. Ganske and McDonald stating that if $R\subseteq S$ are finite local rings such that $S$ is a separable $R$-algebra, then $R\subseteq S$ is a Galois ring extension. We identify the homological error in the published proof of that assertion. Let $(A,M)$ be a finite special principal ideal ring (SPIR), but not a field, such that $M$ has index of nilpotency $\alpha$ ($\ge 2$). Impose the uniform distribution on the (finite) set of ($A$-algebra) isomorphism classes of the minimal ring extensions of $A$. If $2\in M$ (for instance, if $A\cong \mathbb{Z}/2^{\alpha }\mathbb{Z}$), the probability that a random isomorphism class consists of ramified extensions of $A$ is at least $2/3$; if $2\notin M$ (for instance, if $A\cong \mathbb{Z}/p^{\alpha }\mathbb{Z}$ for some odd prime $p$), the corresponding probability is at least $3/4$. Additional applications, examples and historical remarks are given.

Let $R$ be a commutative ring with nonzero identity. The annihilator graph of $R$, denoted by $AG(R)$, is the (undirected) graph whose vertex set is the set of all nonzero zero-divisors of $R$ and two distinct vertices $x$ and $y$ are adjacent if and only if ${\mathrm{\text{ann}}}_R(xy)\ne {\mathrm{\text{ann}}}_R(x)\cup {\mathrm{\text{ann}}}_R(y)$. In this paper, we study the metric dimension of annihilator graphs associated with commutative rings and some metric dimension formulae for annihilator graphs are given.

By modifying the existing definition of a compressed zero-divisor graph ${\mathrm{\Gamma }}_E(K)$, we define a compressed zero-divisor graph $\mathrm{\Theta }(K)$ of a finite commutative unital ring $K$, where the compression is performed by means of the associatedness relation (a refinement of the relation used in the definition of ${\mathrm{\Gamma }}_E(K)$). We prove that this is the best possible compression which induces a functor $\mathrm{\Theta }$, and that this functor preserves categorial products (in both directions). We use the structure of $\mathrm{\Theta }(K)$ to characterize important classes of finite commutative unital rings, such as local rings and principal ideal rings.

If $A\subseteq B$ are (commutative) rings, $[A,B]$ denotes the set of intermediate rings and $(A,B)$ is called an almost valuation (AV)-ring pair if each element of $[A,B]$ is an AV-ring. Many results on AV-domains and their pairs are generalized to the ring-theoretic setting. Let $R\subseteq S$ be rings, with ${\overline{R}}_S$ denoting the integral closure of $R$ in $S$. Then $(R,S)$ is an AV-ring pair if and only if both $(R,{\overline{R}}_S)$ and $({\overline{R}}_S,S)$ are AV-ring pairs. Characterizations are given for the AV-ring pairs arising from integrally closed (respectively, integral; respectively, minimal) ring extensions $R\subseteq S$. If $(R,S)$ is an AV-ring pair, then $R\subseteq S$ is a P-extension. The AV-ring pairs $(R,S)$ arising from root extensions are studied extensively. Transfer results for the “AV-ring” property are obtained for pullbacks of $(B,I,D)$ type, with applications to pseudo-valuation domains, integral minimal ring extensions, and integrally closed maximal non-AV subrings. Several sufficient conditions are given for $(R,S)$ being an AV-ring pair to entail that $S$ is an overring of $R$, but there exist domain-theoretic counter-examples to such a conclusion in general. If $(R,S)$ is an AV-ring pair and $R\subseteq S$ satisfies FCP, then each intermediate ring either contains or is contained in ${\overline{R}}_S$. While all AV-rings are quasi-local going-down rings, examples in positive characteristic show that an AV-domain need not be a divided domain or a universally going-down domain.

A bounded poset $P:=(P,0,1,\le )$ is said to be lower finite if $P$ is infinite and for all $1\ne x\in P$, there are but finitely many $y\in P$ such that $y\le x.$ In this paper, we classify the modules $M$ over a commutative ring $R$ with identity with the property that the lattice $L_R(M)$ of $R$-submodules $M$ (under set-theoretic containment) is lower finite. Our results are summarized in Theorem 3.1 at the end of this note.

The main purpose of this paper is to answer a question which was left open in N. Jarboui and S. Aljubran [Maximal non-integrally closed subrings of an integral domain, Ric. Mat. (2020), doi.org/10.1007/s11587-020-00500-0] asking for a characterization of maximal non-integrally closed subrings of arbitrary rings.

Let $R$ be a commutative ring with identity. The prime ideal sum graph of $R$, denoted by $\mathrm{\text{PIS}}(R)$, is a graph whose vertices are nonzero proper ideals of $R$ and two distinct vertices $I$ and $J$ are adjacent if and only if $I+J$ is a prime ideal of $R$. In this paper, we study some connections between the graph-theoretic properties of this graph and some algebraic properties of rings. The clique number, the chromatic number and the domination number of the prime ideal sum graph for some classes of rings are studied. It is observed that under which condition $PIS(R)$ is complete. Moreover, the diameter and the girth of $\mathrm{\text{PIS}}(R)$ are studied.

Let $R$ be a finite commutative ring and $Z_N(R)=\{x\in R:xy$ is nilpotent for some $0\ne y\in R\}$. The nilpotent graph${\mathrm{\Gamma }}_N(R)$ of $R$ is the simple undirected graph with vertex set ${Z_N(R)}^{\ast }=Z_N(R)\setminus \{0\}$ in which two vertices $x$ and $y$ are adjacent if and only if $xy$ is nilpotent. In this paper, we observe a relationship between zerodivisor graphs, essential graphs and nilpotent graphs of $R$. In continuation of genus characterizations in [T. Asir K. Mano and T. Tamizh Chelvam, Correction to: Classification of non-local rings with genus two zerodivisor graphs, Soft Comput.25 (2021) 3355–3356; K. Selvakumar and M. Subajini, Finite commutative ring with genus two essential 10 graph, J. Algebra Appl.16(2) (2018) 1850121], we classify all finite commutative rings (up to isomorphism) whose nilpotent graphs are of genus two.

Let $p$ be an odd prime number and $A:=\mathbb{Z}/p^2\mathbb{Z}$. Assume that either $($i$)$$p=3$ or $($ii$)$$p\ge 5$ and $p$ is congruent to either $5$ or $11$ modulo $12$$($respectively, assume that $p\ge 5$ and $p$ is congruent to either $1$ or $7$ modulo $12)$. Then there exist exactly five $($respectively, exactly seven$)$ isomorphism classes of rings $R$ for which there exists a tower $A\subset {ℬ}\subset R$ of ramified $($integral minimal$)$ ring extensions such that ${ℬ}$ is the only ring properly contained between $A$ and $R$. We produce a set of isomorphism class representatives of such $R$ and, for each such $R$, we show that, up to isomorphism, the corresponding ring ${ℬ}$ is the idealization $A(+){𝔽}_p$. One consequence, for each integer $n>1$ whose prime-power factorization $n=\prod p_i^{e_i}$ (with pairwise distinct prime numbers $p_i$ and positive integers $e_i$) satisfies $e_i\le 4$ for all $i$, is a classification, up to isomorphism, of the rings that have cardinality $n$ and exactly two proper (unital) subrings.