Some results on the ideal-based cozero-divisor graph of a commutative ring

TL;DR

The paper addresses the structure and planarity of the ideal-based cozero-divisor graph $Γ''_I(R)$, introducing it as a dual/generalization of related graphs and examining its behavior under direct products. It shows that for R = R1 × R2 with I = I1 × I2, adjacencies are componentwise, yielding a connected graph of diameter at most 3 and giving lower bounds on ω and χ from the factor graphs; it derives planarity consequences and provides a finite classification of planar Γ''_I(R) for non-local finite rings, with additional local-ring criteria and ara(m/I) bounds guiding planarity in local cases. The results connect ideal-based cozero-divisor graphs with classical planarity theory (Kuratowski) and offer concrete classifications for finite rings, enriching the understanding of how ring structure governs cozero-divisor graph properties.

Abstract

Let R be a commutative ring with identity and I be an ideal of R. The cozero-divisor graph with respect to I, denoted by $Γ''_I(R)$, is the graph of R with vertices {x \in R -I :xR +I \not=R} and two distinct vertices $x$ and $y$ are adjacent if and only if $x \not \in yR+I$ and $y \not \in xR+I$.In this paper, we obtained some results on $Γ''_I(R)$.

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Theorems & Definitions (46)

• Lemma 2.1
• proof
• Corollary 2.2
• proof
• Lemma 2.3
• proof
• Corollary 2.4
• proof
• Corollary 2.5
• proof