Ideal-based quasi cozero divisor graph of a commutative ring

TL;DR

This paper introduces and studies a new ideal-based quasi cozero divisor graph of a commutative ring, exploring its properties and relationships with existing zero-divisor and cozero-divisor graphs.

Contribution

It defines a novel graph structure $Q ext{Gamma}''_I(R)$ related to ideals in commutative rings and investigates its properties and connections to existing graphs.

Findings

Characterization of the graph's structure.

Relationships with zero-divisor and cozero-divisor graphs.

Conditions for connectivity and other graph properties.

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 $\Gamma''_I(R)$, is the graph of $R$ with vertices $\{x \in R \setminus I | xR + I \neq R\}$, 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 $Q\Gamma''_I(R)$ of R with vertices $\{x \in R \setminus \sqrt{I} | xR + I \neq R$ and $xR + \sqrt{I} = xR + I\}$ and two distinct vertices x and y are adjacent if and only if $x \notin yR + I$ and $y \notin xR + I$.