Yazd University Algebraic Structures and Their Applications 2382-9761 12 4 2025 11 01 The strongly annihilating-ideal graph of a commutative ring with respect to an ideal 353 364 3757 10.22034/as.2025.20379.1661 EN Zahra Mahmudiankoruie Department of Mathematics, Faculty of Science, University of Qom, Qom, Iran Mohammad Hasan Naderi Department of Mathematics, Faculty of Science, University of Qom, Qom, Iran Journal Article 2023 07 26 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 ${\rm SAG}(R/I)$.<br />  https://as.yazd.ac.ir/article_3757_5d55a8281635beee6da4fbc84e014b10.pdf