THE ANNIHILATOR IDEAL GRAPH OF A COMMUTATIVE RING

- Journal title : Journal of the Korean Mathematical Society
- Volume 52, Issue 2, 2015, pp.417-429
- Publisher : The Korean Mathematical Society
- DOI : 10.4134/JKMS.2015.52.2.417

Title & Authors

THE ANNIHILATOR IDEAL GRAPH OF A COMMUTATIVE RING

Alibemani, Abolfazl; Bakhtyiari, Moharram; Nikandish, Reza; Nikmehr, Mohammad Javad;

Alibemani, Abolfazl; Bakhtyiari, Moharram; Nikandish, Reza; Nikmehr, Mohammad Javad;

Abstract

Let R be a commutative ring with unity. The annihilator ideal graph of R, denoted by , is a graph whose vertices are all non-trivial ideals of R and two distinct vertices I and J are adjacent if and only if or . In this paper, we study some connections between the graph-theoretic properties of this graph and some algebraic properties of rings. We characterize all rings whose annihilator ideal graphs are totally disconnected. Also, we study diameter, girth, clique number and chromatic number of this graph. Moreover, we study some relations between annihilator ideal graph and zero-divisor graph associated with R. Among other results, it is proved that for a Noetherian ring R if is triangle free, then R is Gorenstein.

Keywords

annihilator ideal graph;diameter;Clique number;

Language

English

Cited by

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