THE ANNIHILATING-IDEAL GRAPH OF A RING

- Journal title : Journal of the Korean Mathematical Society
- Volume 52, Issue 6, 2015, pp.1323-1336
- Publisher : The Korean Mathematical Society
- DOI : 10.4134/JKMS.2015.52.6.1323

Title & Authors

THE ANNIHILATING-IDEAL GRAPH OF A RING

ALINIAEIFARD, FARID; BEHBOODI, MAHMOOD; LI, YUANLIN;

ALINIAEIFARD, FARID; BEHBOODI, MAHMOOD; LI, YUANLIN;

Abstract

Let S be a semigroup with 0 and R be a ring with 1. We extend the definition of the zero-divisor graphs of commutative semigroups to not necessarily commutative semigroups. We define an annihilating-ideal graph of a ring as a special type of zero-divisor graph of a semigroup. We introduce two ways to define the zero-divisor graphs of semigroups. The first definition gives a directed graph (S), and the other definition yields an undirected graph (S). It is shown that (S) is not necessarily connected, but (S) is always connected and diam. For a ring R define a directed graph to be equal to , where is a semigroup consisting of all products of two one-sided ideals of R, and define an undirected graph to be equal to . We show that R is an Artinian (resp., Noetherian) ring if and only if has DCC (resp., ACC) on some special subset of its vertices. Also, it is shown that is a complete graph if and only if either is a direct product of two division rings, or R is a local ring with maximal ideal m such that . Finally, we investigate the diameter and the girth of square matrix rings over commutative rings where .

Keywords

rings;semigroups;zero-divisor graphs;annihilating-ideal graphs;

Language

English

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