The strongly annihilating-ideal graph of a commutative ring with respect to an ideal

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Document Type : Research Paper

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Department of Mathematics, Faculty of Science, University of Qom, Qom, Iran

Abstract

For a commutative ring $R$ with identity, ${\rm SAG}(R)$ be the graph whose vertices are the nonzero annihilating ideals of $R$ and with two distinct nonzero annihilating ideals $I$ and $J$ joined by an edge when $I\cap {\rm Ann}(J)\neq (0)$ and $J\cap {\rm Ann}(I) \neq (0)$. Also, strongly Annihilating-ideal graph with respect to an ideal $(I)$, that it is shown by ${\rm SAG}_I(R)$, is the graph whose vertices are all ideals of $R$ such that $K\not\subseteq I$ and for some ideal $J$ that $J\not\subseteq I$, $KJ \subseteq I$, and distinct vertices $K$ and $J$ are adjacent if and only if $J\cap {\rm Ann}_I(K)\not\subseteq I$ and $K\cap {\rm Ann}_I(J)\not\subseteq I$. In this paper, we study the notion of ${\rm SAG}_I(R)$. Also, among other results, we give some results about the relationships between $\rm{ SAG}_I(R)$ and ${\rm SAG}(R/I)$.
 

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References

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Algebraic Structures and Their Applications

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Available Online from 16 April 2025

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