The Annihilator Ideal Graph of a Commutative Ring

Abstract

Abstract. Let R be a commutative ring with unity. The annihilator ideal graph of R, denoted by Γ Ann (R), 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 I ∩ Ann(J) = {0} or J ∩ Ann(I) = {0}. 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 Γ Ann (R) is triangle free, then R is Gorenstein.

When the annihilator graph of a commutative ring is planar or toroidal?

“…Recently, a major part of research in algebraic combinatorics has been devoted to the application of graph theory and combinatorics in abstract algebra. There are a lot of papers which apply combinatorial methods to obtain algebraic results in ring theory (see [2], [3], [6] and [13]). Moreover, for most recent study in this field see [7] and [14].…”

When the Annihilator Graph of a Commutative Ring Is Planar or Toroidal?

“…Recently, a major part of research in algebraic combinatorics has been devoted to the application of graph theory and combinatorics in abstract algebra. There are a lot of papers which apply combinatorial methods to obtain algebraic results in ring theory (see [2], [3], [6] and [13]). Moreover, for most recent study in this field see [7] and [16].…”