On $S$-$(\delta, 2)$-primary ideals of a commutative ring

summary: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.