THE ANNIHILATOR IDEAL GRAPH OF A COMMUTATIVE RING
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Let R be a commutative ring with unity. The annihilator ideal graph of R, denoted by ΓAnn(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 diame- ter, 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 ΓAnn(R) is triangle free, then R is Gorenstein.
Abstract
Let R be a commutative ring with unity. The annihilator ideal graph of R, denoted by ΓAnn(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 diame- ter, 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 ΓAnn(R) is triangle free, then R is Gorenstein.
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