Received September 14, 2025. Published online February 17, 2026.
Abstract: Let $R$ be a commutative ring with identity, $S$ be a multiplicative set of $R$, ${\rm Id}(R)$ be the set of all ideals of $R$, and $\delta\colon{\rm Id}(R) \rightarrow{\rm Id}(R)$ be a function. Then $\delta$ is called an expansion function of ideals of $R$ if whenever $L$, $I$, $J$ are ideals of $R$ with $J \subseteq I$, we have $L \subseteq\delta(L)$ and $\delta(J)\subseteq\delta(I)$. Let $\delta$ be an expansion function of ideals of $R$. We introduce the concept of $S$-$(\delta, 2)$-primary ideal which is a generalization of $(\delta,2)$-primary ideal. Let $P$ be a proper ideal of $R$ disjoint with $S$. We say that $P$ is an $S$-$(\delta, 2)$-primary ideal of $R$ if there exists $s \in S$ such that for all $a,b \in R$, if $ab \in P$, then $sa^2 \in P$ or $sb^2 \in\delta(P)$. We next study the possible transfer of the above ideal property to the direct product of rings, quotient rings, localizations, trivial ring extensions, and amalgamation rings along an ideal.
Affiliations: Chahrazade Bakkari, Rachid Hachache, Department of Mathematics, Faculty of Science, University Moulay Ismail Meknes, BP 298, Avenue Zitoune, Meknes, Morocco, e-mail: cbakkari@hotmail.com, rachid.hachache@gmail.com; Suat Koç, Department of Mathematics, Marmara University, Küçük Çamlica, 34718 Istanbul, Turkey, e-mail: suat.koc@marmara.edu.tr; Najib Mahdou, Laboratory of Modelling and Mathematical Structures, Department of Mathematics, Faculty of Science and Technology of Fez, Box 2202, University S.M. Ben Abdellah Fez, Morocco, e-mail: mahdou@hotmail.com; Ünsal Tekir, Department of Mathematics, Marmara University, Ziverbey-Kadiköy, 34722 Istanbul, Turkey, e-mail: utekir@marmara.edu.tr; Violeta Leoreanu-Fotea (corresponding author), Faculty of Mathematics, Al. I. Cuza University of Iaşi, Bd Carol I, 11, Romania, e-mail: violeta.fotea@uaic.ro
