Hung-Jen Chiang-Hsieh, Neal O. Smith, Hsin-Ju Wang
Let $R$ be a commutative ring and $Γ(R)$ denote its zero-divisor graph. In this paper, we investigate the genus number of the compact Riemann surface which $Γ(R)$ can be embedded and illustrate all finite commutative rings $R$ (up to isomorphism) such that $Γ(R)$ is either toroidal or planar.
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In this paper, we continue the program initiated by I. Beck's now classical paper concerning zero-divisor graphs of commutative rings. After the success of much research regarding zero-divisor graphs, many authors have turned their attention to studying divisor graphs of non-zero elements in the ring, the so called irreducible divisor graph. In this paper, we construct several different associated irreducible divisor graphs of a commutative ring with unity using various choices for the definition of irreducible and atomic in the literature. We continue pursuing the program of exploiting the in…
Partitioning zero-divisor graphs of finite commutative rings into global defensive alliances
本文研究将有限交换环的零因子图顶点集划分为全局防御联盟的问题，刻画了满足该性质的多种有限交换环，并给出示例说明结果的适用范围与局限。
On the vertex connectivity of weakly zero-divisor graph of commutative rings
本文研究交换环的弱零因子图WΓ(R)，证明当R为阿廷环或既约环时，WΓ(R)的顶点连通性等于其最小度，并刻画了该图的所有最小度顶点。
Let $R$ be a commutative ring and let $I$ be an ideal of $R$. In this paper, we introduce the cozero-divisor graph $\acuteΓ_I(R)$ of $R$ and obtain some related results.
The Unit-Zero Divisor Graph of a Commutative Ring
本文提出交换环的单位-零因子图$G_{UZ}(R)$这一新关联框架，定义其顶点与邻接规则，研究其正则性等结构性质及环代数结构对这些性质的影响。
Much work has been done on generalized factorization techniques in integral domains, namely $τ$-factorization. There has also been substantial progress made in investigating factorization in commutative rings with zero-divisors. This paper seeks to synthesize work done in these two areas and extend the notion of $τ$-factorization to commutative rings that need not be domains. In addition, we look into particular types of $τ$ relations, which are interesting when there are zero-divisors present. We then proceed to classify commutative rings that satisfy the finite factorization properties given…
Much work has been done on generalized factorization techniques in integral domains, namely $τ$-factorization. There has also been substantial progress made in investigating factorization in commutative rings with zero-divisors. There are many ways authors have decided to study factorization when zero-divisors present. This paper focuses on the method $τ$-complete factorizations developed by D.D. Anderson and A. Frazier. There is a natural way to extend $τ$-complete factorization to commutative rings with zero-divisors. $τ$-complete factorization is a natural way to think of refining factoriza…
Recently substantial progress has been made on generalized factorization techniques in integral domains, in particular $τ$-factorization. There has also been advances made in investigating factorization in commutative rings with zero-divisors. One approach which has been found to be very successful is that of U-factorization introduced by C.R. Fletcher. We seek to synthesize work done in these two areas by generalizing $τ$-factorization to rings with zero-divisors by using the notion of U-factorization.
In this article we revisit a problem regarding Bezout domains, namely, whether every Bezout domain is an elementary divisor domain. We prove that a Bezout domain in which every maximal ideal is principal is an elementary divisor ring
Let $R$ be a commutative ring with unity and $R^{+}$ be $Z^*(R)$ be the additive group and the set of all non-zero zero-divisors of $R$, respectively. We denote by $\mathbb{CAY}(R)$ the Cayley graph $Cay(R^+,Z^*(R))$. In this paper, we study $\mathbb{CAY}(R)$. Among other results, it is shown that for every zero-dimensional non-local ring $R$, $\mathbb{CAY}(R)$ is a connected graph of diameter 2. Moreover, for a finite ring $R$, we obtain the vertex connectivity and the edge connectivity of $\mathbb{CAY}(R)$. We investigate rings $R$ with perfect $\mathbb{CAY}(R)$ as well. We also study $Reg(\…
In this paper, we study divisorial ideals of a Hibi ring which is a toric ring arising from a partially ordered set. We especially characterize the special class of divisorial ideals called conic using the associated partially ordered set. Using our description of conic divisorial ideals, we also construct a module giving a non-commutative crepant resolution (= NCCR) of the Segre product of polynomial rings. Furthermore, applying the operation called mutation, we give other modules giving NCCRs of it.
For a commutative semigroup $S$ with 0, the zero-divisor graph of $S$ denoted by $Γ(S)$ is the graph whose vertices are nonzero zero-divisor of $S$, and two vertices $x$, $y$ are adjacent in case $xy=0$ in $S$. In this paper we study the case where the graph $Γ(S)$ is complete $r$-partite for a positive integer $r$. Also we study the commutative semigroups which are finitely colorable.
Recently there has been a flurry of research on generalized factorization techniques in both integral domains and rings with zero-divisors, namely $τ$-factorization. There are several ways that authors have studied factorization in rings with zero-divisors. This paper focuses on the method of regular factorizations introduced by D.D. Anderson and S. Valdes-Leon. We investigate how one can extend the notion of $τ$-factorization to commutative rings with zero-divisors by using the regular factorization approach. The study of regular factorization is particularly effective because the distinct no…
The global defensive $k$-alliance is a very well studied notion in graph theory, it provides a method of classification of graphs based on relations between members of a particular set of vertices. In this paper we explore this notion in zero-divisor graph of commutative rings. The established results generalize and improve recent work by Muthana and Mamouni who treated a particular case for $k=-1$ known by the global defensive alliance. Various examples are also provided which illustrate and delimit the scope of the established results.
We investigate eigenvalues of the zero-divisor graph $Γ(R)$ of finite commutative rings $R$ and study the interplay between these eigenvalues, the ring-theoretic properties of $R$ and the graph-theoretic properties of $Γ(R)$. The graph $Γ(R)$ is defined as the graph with vertex set consisting of all non-zero zero-divisors of $R$ and adjacent vertices $x,y$ whenever $xy = 0$. We provide formulas for the nullity of $Γ(R)$, i.e. the multiplicity of the eigenvalue 0 of $Γ(R)$. Moreover, we precisely determine the spectra of $Γ(\mathbb Z_p \times \mathbb Z_p \times \mathbb Z_p)$ and $Γ(\mathbb Z_p …
Some results on the ideal-based cozero-divisor graph of a commutative ring
本文研究含单位元交换环R关于理想I的余零因子图Γ''_I(R)，给出了该图的一些性质结果。
A (not necessarily commutative) Krull monoid---as introduced by Wauters---is defined as a completely integrally closed monoid satisfying the ascending chain condition on divisorial two-sided ideals. We study the structure of these Krull monoids, both with ideal theoretic and with divisor theoretic methods. Among others we characterize normalizing Krull monoids by divisor theories. Based on these results we give a criterion for a Krull monoid to be a bounded factorization monoid, and we provide arithmetical finiteness results in case of normalizing Krull monoids with finite Davenport constant.
Characterization of rings with planar, toroidal or projective planar prime ideal sum graphs
该文研究含单位元交换环的素理想和图的图论性质与环代数性质的关联，分类了交叉帽数至多2、亏格为1的非局部交换Artin环，证明不存在素理想和图为射影平面的这类环。
The first goal of the present paper is to study the class groups of the edge rings of complete multipartite graphs, denoted by $\Bbbk[K_{r_1,\ldots,r_n}]$, where $1 \leq r_1 \leq \cdots \leq r_n$. More concretely, we prove that the class group of $\Bbbk[K_{r_1,\ldots,r_n}]$ is isomorphic to $\mathbb{Z}^n$ if $n =3$ with $r_1 \geq 2$ or $n \geq 4$, while it turns out that the excluded cases can be deduced into Hibi rings. The second goal is to investigate the special class of divisorial ideals of $\Bbbk[K_{r_1,\ldots,r_n}]$, called conic divisorial ideals. We describe conic divisorial ideals fo…
Categorial properties of compressed zero-divisor graphs of finite commutative rings
该研究定义了有限交换含幺环基于相伴关系的压缩零因子图$\u0398(K)$，证明其为诱导函子的最优压缩且函子保持范畴积，并借助该图结构刻画了局部环、主理想环等重要环类。
On transfer homomorphisms in commutative rings with zero-divisors
本文研究含零因子交换环中正则元素幺半群的算术，给出子环包含映射成为转移同态的充分条件，从而将 Krull 幺半群的算术性质转移到子环。
Let $R$ be a commutative ring with unity not equal to zero and let $Γ(R)$ be a zero-divisor graph realized by $R$. For a simple, undirected, connected graph $G = (V, E)$, a {\it total perfect code} denoted by $C(G)$ in $G$ is a subset $C(G) \subseteq V(G)$ such that $|N(v) \cap C(G)| = 1$ for all $v \in V(G)$, where $N(v)$ denotes the open neighbourhood of a vertex $v$ in $G$. In this paper, we study total perfect codes in graphs which are realized as zero-divisor graphs. We show a zero-divisor graph realized by a local commutative ring with unity admits a total perfect code if and only if the…
Counter-example to Conjectures on Complemented Zero-Divisor Graphs of Semigroups
本文构造了一个带零元的交换半群，作为Bender等人2024年提出的关于补零因子图的两个猜想的反例，证明其不成立。
The $i$-extended ideal-based cozero-divisor graph of a commutative ring
本文研究了交换环的i-扩展理想基余零除数图，定义了图的顶点和边的条件，并探讨了其结构性质。
A Graph-Theoretic Approach to Ring Analysis: Dominant Metric Dimensions in Zero-Divisor Graphs
本文通过图论方法研究有限交换环的零因子图，建立了主导度量维数的一般界，并分析了特定环类，揭示了图结构参数与代数性质的联系。
In 1988, I.Beck showed that the chromatic number of G(Zn) is equal to its clique number.In 2004, S.Akbari and A.Mohammadian proved that the edge chromatic number of G(Zn) is equal to its maximum degree,in 2008, J.Skowronek-kaziow give formulas calculating the clique number and the maximum degree of G(Zn), but he have a error about clique number of G(Zn), we consider the zero-divisor graph G(Zn) of the ring Zn.we give formulas calculating the clique number of G(Zn).We present a constructed method to calculate the clique number.
We determine the automorphism group of the zero-divisor digraph of the semiring of matrices over an antinegative commutative semiring with a finite number of zero-divisors.
Using the concept of ring of Gelfand range 1 we proved that a commutative Bezout domain is an elementary divisor ring iff it is a ring of Gelfand range 1. Obtained results give a solution of problem of elementary divisor rings for different classes of commutative Bezout domains, in particular, $PM^{*}$, local Gelfand domains and so on.
In this paper, we introduce a new graph whose vertices are the nonzero zero-divisors of commutative ring $R$ and for distincts elements $x$ and $y$ in the set $Z(R)^{\star}$ of the nonzero zero-divisors of $R$, $x$ and $y$ are adjacent if and only if $xy=0$ or $x+y\in Z(R)$. we present some properties and examples of this graph and we study his relation with the zero-divisor graph and with a subgraph of total graph of a commutative ring.
We characterize the commutative rings whose ideals (resp. regular ideals) are products of radical ideals.
The Homology Groups of Zero Divisor Graphs of Finite Commutative Rings
本文研究了有限交换环零因子图的同调群，推导了相关复形的 Kunneth 公式，并给出了 Cohen-Macaulay 性质的分类及部分拓扑空间无法实现的结论。
Some factorization properties of idealization in commutative rings with zero divisors
本文研究交换环中模的理想化的分解性质，证明 ACCP 的等价条件，说明 BF 性质不一定保持并给出保持条件，同时刻画了理想化环为 UFR 的情形。
In 1988, I. Beck introduced the notion of a zero-divisor graph of a commutative rings with $1$. There have been several generalizations in recent years. In particular, in 2007 J. Coykendall and J. Maney developed the irreducible divisor graph. Much work has been done on generalized factorization, especially $τ$-factorization. The goal of this paper is to synthesize the notions of $τ$-factorization and irreducible divisor graphs in domains. We will define a $τ$-irreducible divisor graph for non-zero non-unit elements of a domain. We show that by studying $τ$-irreducible divisor graphs, we find …
Given a certain factorization property of a ring $R$, we can ask if this property extends to the polynomial ring over $R$ or vice versa. For example, it is well known that $R$ is a unique factorization domain if and only if $R[X]$ is a unique factorization domain. If $R$ is not a domain, this is no longer true. In this paper we survey unique factorization in commutative rings with zero divisors, and characterize when a polynomial ring over an arbitrary commutative ring has unique factorization.
Toroidal involutory Cayley graphs
本文研究含单位元有限交换环的对合Cayley图，对所有该图为环面图的环R完成了分类，丰富了Cayley图的拓扑嵌入相关结果。
We study the zero divisor graph determined by equivalence classes of zero divisors of a commutative Noetherian ring R. We demonstrate how to recover information about R from this structure. In particular, we determine how to identify associated primes from the graph.
In this paper, we determine bipartite graphs and complete graphs with horns, which are realizable as zero-divisor graphs of po-semirings. As applications, we classify commutative rings $R$ whose annihilating-ideal graph $\mathbb {AG}(R)$ are either bipartite graphs or complete graphs with horns.
Betti numbers for cochordal zero-divisor graphs of commutative rings
本文研究了几类有限链环家族的余除数图，通过 cochordal 构造系统计算其边理想同调不变量。证明了C(q,L)为cochordal图，修正并改进了其边理想Betti数公式，并针对Gaussian商环和截断多项式环进行专门研究。
In this paper, we characterize chordal and perfect zero-divisor graphs of finite posets. Also, it is proved that the zero-divisor graphs of finite posets and the complement of zero-divisor graphs of finite $0$-distributive posets satisfy the Total Coloring Conjecture. These results are applied to the zero-divisor graphs of finite reduced rings, the comaximal ideal graph of rings, the annihilating ideal graphs, the intersection graphs of ideals of rings, and the intersection graphs of subgroups of cyclic groups. In fact, it is proved that these graphs associated with a commutative ring $R$ with…
The Weak Zero-Divisor Difference Graph of a Finite Commutative Ring
研究有限交换环的弱零因子差图\(\DR\)，通过引入该图并为有限约化环发展结构理论，确定其在不同条件下的性质，如连通性、完美性等，还给出相关参数的封闭形式及重构定理，指出约化环假设不可轻易去掉。
Let $M$ be an $R$-module and $c$ the function from $M$ to the ideals of $R$ defined by $c(x) = \cap \lbrace I \colon I \text{is an ideal of} R \text{and} x \in IM \rbrace $. $M$ is said to be a content $R$-module if $x \in c(x)M $, for all $x \in M$. $B$ is called a content $R$-algebra, if it is a faithfully flat and content $R$-module and it satisfies the Dedekind-Mertens content formula. In this article, we prove some new results for content modules and algebras by using ideal theoretic methods.
Ideal-based quasi cozero divisor graph of a commutative ring
本文在有单位元的交换环中引入并研究了一个基于理想的拟非零因子无向图$QΓ''_I(R)$，通过结合根理想$\sqrt{I}$与理想I的条件重新定义了图的顶点集与邻接关系。
On maximal common divisors in Puiseux monoids
本文研究了 Puiseux monoids 中极大公因子的存在性，并建立了其与原子性及主理想升链条件之间的联系。
In this paper, we show that it is possible for a commutative ring with identity to be non-atomic (that is, there exist non-zero nonunits that cannot be factored into irreducibles) and yet have a strongly atomic polynomial extension. In particular, we produce a commutative ring with identity, R, that is antimatter (that is, R has no irreducibles whatsoever) such that R[t] is strongly atomic. What is more, given any nonzero nonunit f(t) in R[t] then there is a factorization of f(t) into irreducibles of length no more than deg(f(t)) + 2.
Rings whose associated extended zero-divisor graphs are complemented
本文继续研究带有恒等元的交换环的扩展零因子图何时为补图和唯一补图，给出了有限环扩展零因子图为补图的完整刻画，并利用环直积与模理想化给出实例。
It is proven that every commutative semihereditary Bezout ring in which any regular element is Gelfand (adequate), is an elementary divisor ring.
Boxicity and Threshold Dimension of Zero Divisor Graphs
该研究针对约化环和主理想整环的商环两类有限交换环，借助新提出的积分覆盖图组合工具，确定了其零因子图的盒维数与阈值维数，解答了相关领域学者近期提出的两个公开问题。
Embeddings of the line graphs associated with the essential graphs of commutative rings
本文研究有限交换环本质图的线图在可定向与不可定向曲面上的最小嵌入，完整分类了线图为平面、外平面或亏格及交叉帽数至多为二的环，并刻画了零因子图线图外平面的非局部环。
The Zero divisor Graph of a commutative ring $R$, denoted by $Γ[R]$, is a graph whose vertices are non-zero zero divisors of $R$ and two vertices are adjacent if their product is zero. In this paper, we consider the zero divisor graph $Γ[\mathbb{Z}_n]$ for $n=p^3$ and $n=p^2q$ with $p$ and $q$ primes. We discuss the adjacency matrix and eigenvalues of the zero divisor graph $Γ[\mathbb{Z}_n]$. We also calculate the energy of the graph $Γ[\mathbb{Z}_n]$.
Rings are $χ$-bounded
本文证明任意环的零因子图在团数有限时满足色数受团数函数控制，巩固了 Beck 猜想，并给出下界，同时指出半环和非结合环情形不成立。
Non-commutative crepant resolutions of toric singularities with divisor class group of rank one
本文证明并分类了除子类群秩一的Gorenstein环状奇点的环状非交换凝聚解消（NCCR），通过上集与Iyama-Wemyss突变建立导出等价，并在非金字塔情形下给出二聚体模型实现。
Let S be a commutative ring, x, y $\in$ S a pair of exact zero divisors, and R = S/(x). Let F be a complex of free R-modules. In this paper we explicitly compute cohomological operators of R over S by constructing endomorphisms of F. We consider some properties of these cohomological operators, as well as provide an example in which these cohomological operators act non-trivially.
Let $R$ be a commutative ring with unity and $Z(R)$ and ${\rm Reg}(R)$ be the set of zero-divisors and non-zero zero-divisors of $R$, respectively. We denote by $T(Γ(R))$, the total graph of $R$, a simple graph with the vertex set $R$ and two distinct vertices $x$ and $y$ are adjacent if and only if $x+y\in Z(R)$. The induced subgraphs on $Z(R)$ and ${\rm Reg}(R)$ are denoted by $Z(Γ(R))$ and $Reg(Γ(R))$, respectively. These graphs were first introduced by D.F. Anderson and A. Badawi in 2008. In this paper, we prove the following result: let $R$ be a finite ring and one of the following condit…
Generalized divisor topology of commutative rings
本文将因子拓扑推广到任意交换环，引入广义因子拓扑，刻画其拓扑性质并揭示与主理想根式序结构的关系。
If $i:A\subset B$ is a commutative ring extension, we show that the group $\mathcal I(A,B)$ of invertible $A$-submodules of $B$ is contracted in the sense of Bass, with $L\mathcal I(A,B)=H^0_{et}(A,i_*\mathbb Z/\mathbb Z)$. This gives a canonical decomposition for $\mathcal I(A[t,\frac1t],B[t,\frac1t])$.
Let $M$ be an $R$-module and $S$ a semigroup. Our goal is to discuss zero-divisors of the semigroup module $M[S]$. Particularly we show that if $M$ is an $R$-module and $S$ a commutative, cancellative and torsion-free monoid, then the $R[S]$-module $M[S]$ has few zero-divisors of degree $n$ if and only if the $R$-module $M$ has few zero-divisors of degree $n$ and Property (A).
The extended zero-divisor graph of the amalgamated duplication of a ring along an ideal
本文研究环沿理想的融合复制的扩展零因子图，刻画其与零因子图重合及完全性的条件，并计算直径和围长。
On zero-divisor graph of the ring of Gaussian integers modulo $2^n$
本文研究了模 $2^n$ 的高斯整数环的零因子图结构，通过因子的相伴类确定了图的大小、色数、团数、独立数和匹配数，并求得了相关拓扑指标。
On Divisor Topology of Commutative Rings
该研究针对整环$R$，通过相伴元等价类构造除数拓扑$D(R)$，探讨其与$R$代数性质的联系，证明原子整环的$D(R)$是Baire空间，刻画赋值整环并给出素元无限性的新拓扑证明。
Let $f : A \rightarrow B$ be a ring homomorphism and $J$ be an ideal of $B$. In this paper, we give a characterization of zero divisors of the amalgamation which is a generalization of Maimani's and Yassemi's work (see \cite{y}). Also, we investigate the transfer of Prüfer domain concept to commutative rings with zero divisors in the amalgamation of $A$ with $B$ along $J$ with respect to $f$ (denoted by $A\bowtie^fJ),$ introduced and studied by D'Anna, Finocchiaro and Fontana in 2009 (see \cite{AFF1} and \cite{AFF2}). Our aim is to provide new classes of commutative rings satisfying this prope…
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