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

TL;DR

This paper introduces the i-extended ideal-based cozero-divisor graph of a commutative ring, exploring its properties and structure based on specific algebraic conditions involving ideals and vertices.

Contribution

It defines a new graph associated with a ring and ideal, extending previous concepts to include parameters m, n, and i, and investigates its properties.

Findings

The graph's adjacency depends on algebraic conditions involving powers of elements.

The structure of the graph reflects properties of the ring and ideal.

New insights into the interplay between ring elements and graph theory.

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 $m\leq i$.