On the genus of nil-graph of ideals of commutative rings

Let R be a commutative ring with identity and let Nil(R) be the ideal of all nilpotent elements of R. Let I(R)={I:I is a non-trivial ideal of R and there exists a non-trivial ideal J such that IJ⊆Nil(R)}. The nil-graph of ideals of R is defined as the simple undirected graph AGN(R) whose vertex set is I(R) and two distinct vertices I and J are adjacent if and only if IJ⊆ Nil(R). In this paper, we study the planarity and genus of AGN(R). In particular, we have characterized all commutative Artin rings R for which the genus of AGN(R) is either zero or one.