Ideal-based quasi cozero divisor graph of a commutative ring

Abstract

Let R be a commutative ring with identity, and let I be an ideal of R. The zero-divisor graph of R with respect to I, denoted by ${\Gamma }_I(R)$, is the graph whose vertices are the set $\{x\in R\setminus I|xy\in I$ for some $y\in R\setminus I\}$, where distinct vertices x and y are adjacent if and only if $xy\in I$. The cozero-divisor graph with respect to I, denoted by ${\Gamma }_I^{\prime \prime }(R)$, is the graph of $R$ with vertices $\{x\in R\setminus I|xR+I\ne R\}$, and two distinct vertices x and y are adjacent if and only if $x\notin yR+I$ and $y\notin xR+I$. In this paper, we introduce and investigate an undirected graph $Q{\Gamma }_I^{\prime \prime }(R)$ of R with vertices $\{x\in R\setminus \sqrt{I}|xR+I\ne R$ and $xR+\sqrt{I}=xR+I\}$ and two distinct vertices x and y are adjacent if and only if $x\notin yR+I$ and $y\notin xR+I$.