Characterization of zero-dimensional rings such that the clique number of their annihilating-ideal graphs is at most four

Document Type : Research Paper

Authors

1 Retired Faculty, Department of Mathematics, Saurashtra University, Rajkot, 360005, India.

2 Department of Mathematics, Dr. Subhash University, Junagadh, 362001, India.

Abstract

The rings considered in this article are commutative with identity which are not integral domains. Let $R$ be a ring. An ideal $I$ of $R$ is said to be an annihilating ideal of $R$ if there exists $r\in R\backslash \{0\}$ such that $Ir = (0)$. Let $\mathbb{A}(R)$ denote the set of all annihilating ideals of $R$ and let $\mathbb{A}(R)^{*} = \mathbb{A}(R)\backslash \{(0)\}$. Recall that the annihilating-ideal graph of $R$, denoted by $\mathbb{AG}(R)$, is an undirected graph whose vertex set is $\mathbb{A}(R)^{*}$ and distinct vertices $I$ and $J$ are adjacent in this graph if and only if $IJ = (0)$. The aim of this article is to characterize zero-dimensional rings such that the clique number of their annihilating-ideal graphs is at most four.
 

Keywords

References

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Algebraic Structures and Their Applications
Volume 10, Issue 2
August 2023
Pages 127-154

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