Some Properties of the Nil-Graphs of Ideals of Commutative Rings

2016/11/10 by R. Nikandish, Nikandish, R., Farzad Shaveisi +1
Mathematics · #Rings, Modules, and Algebras #Advanced Topics in Algebra #Commutative Algebra and Its Applications

paper · pdf · doi:10.48550/arxiv.1611.03730

Abstract

Let R be a commutative ring with identity and \rm Nil(R) be the set of nilpotent elements of R. The nil-graph of ideals of R is defined as the graph \mathbbAG_N(R) whose vertex set is \I: (0)≠ I\lhd R and there exists a non-trivial ideal J such that IJ⊆ \rm Nil(R)\ and two distinct vertices I and J are adjacent if and only if IJ⊆ \rm Nil(R). Here, we study conditions under which \mathbbAG_N(R) is complete or bipartite. Also, the independence number of \mathbbAG_N(R) is determined, where R is a reduced ring. Finally, we classify Artinian rings whose nil-graphs of ideals have genus at most one.

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