The $i$-extended ideal-based cozero-divisor graph of a commutative ring

Abstract

Let R be a commutative ring with identity and let J be an ideal of R. In this paper, we introduce and investigate the notion of the i-extended ideal-based cozero-divisor graph of R. This graph, denoted by ${\overline{{\Gamma }^{\prime \prime }}}_{Ji}(R)$, is a simple graph of R whose vertex set is $x\in RJ:xR+J\ne R$. Two distinct vertices $x$ and $y$ are adjacent if and only if $x^m\notin y^{n}R+J$ and $y^n\notin x^{m}R+J$ for some positive integers m and n with $n\le i$ and $m\le i$.