A GENERALIZATION OF THE ZERO-DIVISOR GRAPH FOR MODULES

- Journal title : Journal of the Korean Mathematical Society
- Volume 51, Issue 1, 2014, pp.87-98
- Publisher : The Korean Mathematical Society
- DOI : 10.4134/JKMS.2014.51.1.087

Title & Authors

A GENERALIZATION OF THE ZERO-DIVISOR GRAPH FOR MODULES

Safaeeyan, Saeed; Baziar, Mohammad; Momtahan, Ehsan;

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 , such that when M = R, 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 in the present article. We show that is connected with . We also show that for a reduced module M with , if and only if is a star graph. Furthermore, we show that for a finitely generated semisimple R-module M such that its homogeneous components are simple, are adjacent if and only if . Among other things, it is also observed that if and only if M is uniform, ann(M) is a radical ideal, and , if and only if ann(M) is prime and .

Keywords

module;zero-divisor graph of modules;girth;diameter;complete bipartite graph;

Language

English

Cited by

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