A GENERALIZATION OF THE ZERO-DIVISOR GRAPH FOR MODULES

Title & Authors
A GENERALIZATION OF THE ZERO-DIVISOR GRAPH FOR MODULES
Safaeeyan, Saeed; Baziar, Mohammad; Momtahan, Ehsan;

Abstract
Let R be a commutative ring with identity and M an R-module. In this paper, we associate a graph to M, say $\small{{\Gamma}(M)}$, such that when M = R, $\small{{\Gamma}(M)}$ is exactly the classic zero-divisor graph. Many well-known results by D. F. Anderson and P. S. Livingston, in [5], and by D. F. Anderson and S. B. Mulay, in [6], have been generalized for $\small{{\Gamma}(M)}$ in the present article. We show that $\small{{\Gamma}(M)}$ is connected with $\small{diam({\Gamma}(M)){\leq}3}$. We also show that for a reduced module M with $\small{Z(M)^*{\neq}M{\backslash}\{0\}}$, $\small{gr({\Gamma}(M))={\infty}}$ if and only if $\small{{\Gamma}(M)}$ is a star graph. Furthermore, we show that for a finitely generated semisimple R-module M such that its homogeneous components are simple, $\small{x,y{\in}M{\backslash}\{0\}}$ are adjacent if and only if $\small{xR{\cap}yR=(0)}$. Among other things, it is also observed that $\small{{\Gamma}(M)={\emptyset}}$ if and only if M is uniform, ann(M) is a radical ideal, and $\small{Z(M)^*{\neq}M{\backslash}\{0\}}$, if and only if ann(M) is prime and $\small{Z(M)^*{\neq}M{\backslash}\{0\}}$.
Keywords
module;zero-divisor graph of modules;girth;diameter;complete bipartite graph;
Language
English
Cited by
1.
ON GRAPHS ASSOCIATED WITH MODULES OVER COMMUTATIVE RINGS,;;

대한수학회지, 2016. vol.53. 5, pp.1167-1182
1.
A conception of zero-divisor graph for categories of modules, Journal of Algebra and Its Applications, 2016, 15, 01, 1650012
2.
Zero-divisor graphs for modules over integral domains, Journal of Algebra and Its Applications, 2017, 16, 05, 1750087
3.
ON GRAPHS ASSOCIATED WITH MODULES OVER COMMUTATIVE RINGS, Journal of the Korean Mathematical Society, 2016, 53, 5, 1167
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