On line upper ideal relation graphs of rings

Abstract

The upper ideal relation graph $\hat{I}{“}_U(R)$ of a commutative ring $R$ with unity is a simple undirected graph with the set of all non-unit elements of $R$ as a vertex set and two vertices $x$, $y$ are adjacent if and only if the principal ideals $(x)$ and $(y)$ are contained in the principal ideal $(z)$ for some non-unit element $z\in R$. This manuscript characterizes all the Artinian rings $R$ such that the graph $\hat{I}{“}_U(R)$ is a line graph. Moreover, all the Artinian rings $R$ for which $\hat{I}{“}_U(R)$ is the complement of a line graph have been described.

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