On the genus of nil-graph of ideals of commutative rings (original) (raw)

Abstract

Let R be a commutative ring with identity and let Nil(R) be the ideal of all 2 nilpotent elements of R. Let I(R) = {I : I is a non-trivial ideal of R and there exists a 3 non-trivial ideal J such that IJ ⊆ Nil(R)}. The nil-graph of ideals of R is defined as the 4 simple undirected graph AG N (R) whose vertex set is I(R) and two distinct vertices I and 5 J are adjacent if and only if IJ ⊆ Nil(R). In this paper, we study the planarity and genus of 6 AG N (R). In particular, we have characterized all commutative Artin rings R for which the 7 genus of AG N (R) is either zero or one.

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