环分析的图论方法：零因子图中的主导度量维数

环分析的图论方法：压缩零因子图中支配度量维数及其与环结构相互作用的探索

A Graph-Theoretical Approach to Ring Analysis: An Exploration of Dominant Metric Dimension in Compressed Zero Divisor Graphs and Its Interplay with Ring Structures

本文利用压缩零因子图的支配度量维数对环进行分类，建立其维数界，并分析该维数与环性质、直径及围长的关系。

On Irreducible Divisor Graphs in Commutative Rings with Zero-Divisors

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…

Non-commutative Krull monoids: A divisor theoretic approach and their arithmetic

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.

Zero-divisor graphs of nilpotent-free semigroups

We find strong relationships between the zero-divisor graphs of apparently disparate kinds of nilpotent-free semigroups by introducing the notion of an \emph{Armendariz map} between such semigroups, which preserves many graph-theoretic invariants. We use it to give relationships between the zero-divisor graph of a ring, a polynomial ring, and the annihilating-ideal graph. Then we give relationships between the zero-divisor graphs of certain topological spaces (so-called pearled spaces), prime spectra, maximal spectra, tensor-product semigroups, and the semigroup of ideals under addition, obtai…

将有限交换环的零因子图划分为全局防御联盟

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。

Eigenvalues of zero-divisor graphs of finite commutative rings

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 …