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 $\overline{\Gamma''}_{Ji}(R)$, is a simple graph of R whose vertex set is ${x \in R \ J : xR + J \not= R}$. Two distinct vertices $x$ and $y$ are adjacent if and only if $x^m \not \in y^nR+J$ and $y^n \not \in x^mR+J$ for some positive integers m and n with $n\leq i$ and $m\leq i$.