On the genus of annihilator intersection graph of commutative rings

Let R be a commutative ring with unity and A(R) be the set of annihilating-ideals of R. The annihilator intersection graph of R, represented by AIG(R), is an undirected graph with A(R)^* as the vertex set and \mathfrak{M} \sim \mathfrak{N} is an edge of AIG(R) if and only if Ann(\mathfrak{M}\mathfrak{N}) \neq Ann(\mathfrak{M}) \cap Ann(\mathfrak{N}), for distinct vertices \mathfrak{M} and \mathfrak{N} of AIG(R). In this paper, we first defined finite commutative rings whose annihilator intersection graph is isomorphic to various well-known graphs, and then all finite commutative rings with a planar or toroidal annihilator intersection graph were characterized.

Keywords:

Annihilating-ideal graph , Annihilator intersection graph , Genus of a graph , planar graph , zero-divisor graph