More on the annihilator-ideal graph of a commutative ring

Abstract

Let R be a commutative ring with identity and A(R) be the set of ideals of R with non-zero annihilator. The annihilator-ideal graph of R, denoted by A I (R), is a simple graph with the vertex set A(R) * := A(R) \ {(0)}, and two distinct vertices I and J are adjacent if and only if Ann R (IJ) = Ann R (I) ∪ Ann R (J).In this paper, we study the affinity between the annihilator-ideal graph and the annihilating-ideal graph AG(R) (a well-known graph with the same vertices and two distinct vertices I, J are adjacent if and only if IJ = 0) associated with R. All rings whose A I (R) = AG(R) and gr(A I (R)) = 4 are characterized. Among other results, we obtain necessary and sufficient conditions under which A I (R) is a star graph. *

On the essential annihilating-ideal graph of commutative rings

“…In [11], Behboodi et al generalized the zero-divisor graph to ideals by defining the annihilating-ideal graph AG(R), with vertex set is A * (R) and two distinct vertices I 1 and I 2 are adjacent if and only if I 1 I 2 = 0. For more details on annihilating-ideal graph, we refer the reader to see [1,2,3,4,6,12,16].…”