Some remarks on the compressed zero-divisor graph

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• 作者：David F. Anderson^a ; ^{anderson@math.utk.edu" class="auth_mail" title="E-mail the corresponding author} ; John D. LaGrange^b ; ^{lagrangej@lindsey.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：13A15 ; 13A99 ; 05C25 ; 05C99 ; 20M14
• 刊名：Journal of Algebra
• 出版年：2016
• 出版时间：1 February 2016
• 年：2016
• 卷：447
• 期：Complete
• 页码：297-321
• 全文大小：598 K

文摘

Let R   be a commutative ring with 1≠0$1\ne 0$. The zero-divisor graph Γ(R)$\mathrm{\Gamma }(R)$ of R is the (undirected) graph with vertices the nonzero zero-divisors of R, and distinct vertices r and s   are adjacent if and only if rs=0$rs=0$. The relation on R   given by r∼s$r\sim s$ if and only if ann_R(r)=ann_R(s)${\text{ann}}_R(r)={\text{ann}}_R(s)$ is an equivalence relation. The compressed zero-divisor graph Γ_E(R)${\mathrm{\Gamma }}_E(R)$ of R   is the (undirected) graph with vertices the equivalence classes induced by ∼ other than [0] and [1], and distinct vertices [r]$[r]$ and [s]$[s]$ are adjacent if and only if rs=0$rs=0$. Let R_E$R_E$ be the set of equivalence classes for ∼ on R  . Then R_E$R_E$ is a commutative monoid with multiplication [r][s]=[rs]$[r][s]=[rs]$. In this paper, we continue our study of the monoid R_E$R_E$ and the compressed zero-divisor graph Γ_E(R)${\mathrm{\Gamma }}_E(R)$. We consider several equivalence relations on R   and their corresponding graph-theoretic translations to Γ(R)$\mathrm{\Gamma }(R)$. We also show that the girth of Γ_E(R)${\mathrm{\Gamma }}_E(R)$ is three if it contains a cycle and determine the structure of Γ_E(R)${\mathrm{\Gamma }}_E(R)$ when it is acyclic and the monoids R_E$R_E$ when Γ_E(R)${\mathrm{\Gamma }}_E(R)$ is a star graph.