An ideal-based cozero-divisor graph of a commutative ring

Habibollah Ansari-Toroghy,Farideh Mahboobi-Abkenar,Faranak Farshadifar

doi:10.5269/bspm.43261

Habibollah Ansari-Toroghy, Farideh Mahboobi-Abkenar + Show 1 more

Open Access

https://doi.org/10.5269/bspm.43261

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Abstract

Let $R$ be a commutative ring and let $I$ be an ideal of $R$. In this article, we introduce the cozero-divisor graph $\acute{\Gamma}_I(R)$ of $R$ and explore some of its basic properties. This graph can be regarded as a dual notion of an ideal-based zero-divisor graph.

Highlights

I is a second ideal of R if and only if ΓI (R) = ∅
Let R be a non-local ring and I be a proper ideal of R
Let I be a proper ideal of R and ΓI (R) a complete bipartite graph with parts Vi, i = 1, 2

Summary

Introduction

Let R be a non-local ring and I a proper ideal of R such that for every element a ∈ J(R), there exists m ∈ M ax(R) and b ∈ m \ J(R) with a ∈ bR. Let R be a non-local ring and I be a proper ideal of R. Suppose that I = R is a finitely generated ideal of R such that AnnR(I) = 0 and x ∈ V (Γ(R)).

Results

Conclusion