When the annihilator-ideal graph is planar or complete bipartite graph

摘要

Let R be a commutative ring with identity and ??(R) be the set of ideals of R with nonzero annihilator. The annihilator-ideal graph of R, denoted by A?(R), is a simple graph with the vertex set ??(R)? := ??(R){(0)}, and two distinct vertices I and J are adjacent if and only if AnnR(IJ)≠AnnR(I) ∪AnnR(J). In this paper, we present some results on the bipartite, complete bipartite, outer planar and unicyclic of the annihilator-ideal graphs of a commutative ring. Among other results, bipartite annihilator-ideal graphs of rings are characterized. Also, we investigate planarity of the annihilator-ideal graph and classify rings whose annihilator-ideal graph is planar.