On the Planarity of the Regular Digraph of Ideals of Commutative Rings

Abstract

Let \(R\) be a commutative ring. The regular digraph of ideals of \(R\), denoted by \(\mathcal{R}(R)\), is a digraph whose vertex-set is the set of all non-trivial ideals of \(R\) and, for every two distinct vertices \(I\) and \(J\), there is an arc from \(I\) to \(J\), whenever \(I\) contains a non-zero divisor of \(J\). In this paper, we investigate the planarity of \(\mathcal{R}(R)\). We also completely characterize the rings \(R\) such that \(\mathcal{R}(R)\) is a ring graph, and the situations under which the genus of \(\mathcal{R}(R)\) is finite. Moreover, we study the independence number and the girth of \(\mathcal{R}(R)\), and also we find all cases that \(\mathcal{R}(R)\) is bipartite.