A generalization of zero divisor graphs associated to commutative rings

Proceedings of the Indian Academy of Sciences. Mathematical sciences

M AFKHAMI^1¹  A ERFANIAN^2² 

[1] Department of Mathematics, University of Neyshabur, P.O. Box 91136-899, Neyshabur, Iran^1;Department of Pure Mathematics, International Campus of Ferdowsi University of Mashhad, P.O. Box 1159-91775, Mashhad, Iran^2

关键词: Zero divisor graph;    lower triangular matrix;    genus;    complete graph;   

DOI  :  

学科分类：数学（综合）

来源: Indian Academy of Sciences

【 摘 要 】

Let $R$ be a commutative ring with a nonzero identity element. For a natural number $n$, we associate a simple graph, denoted by $\Gamma^{n}_{R}$, with $R^{n}\backslash\{0\}$ as the vertex set and two distinct vertices $X$ and $Y$ in $R^{n}$ being adjacent if and only if there exists an $n\times n$ lower triangular matrix $A$ over $R$ whose entries on the main diagonal are nonzero and one of the entries on the main diagonal is regular such that $X^{T} AY = 0$ or $Y^{T} AX = 0$, where, for a matrix $B$, $B^{T}$ is the matrix transpose of $B$. If $n = 1$, then $\Gamma^{n}_{R}$ is isomorphic to the zero divisor graph $\Gamma(R)$, and so $\Gamma^{n}_{R}$ is a generalization of $\Gamma(R)$ which is called a generalized zero divisor graph of $R$. In this paper, we study some basic properties of $\Gamma^{n}_{R}$. We also determine all isomorphic classes of finite commutative rings whose generalized zero divisor graphs have genus at most three.

【 授权许可】

CC BY