S-r-$ideals in Commutative Rings

This article introduces the notion of $S-r-$ideals in commutative ring $H$ as a generalization of $ r-$ideals. Let $S$ be a multiplicatively closed subset and $A$ an ideal of $H$ with $S\cap A= \emptyset$. $A$ is called $S-r-ideal$ if $ \exists s\in S$ such that for $w,z\in H,\; wz\in A$ and $Ann(w)=0$, then $sz\in A.$ Basic properties of $S-r-$ideals are given. It is shown that an $r-$ideal is always an $S-r-$ideal, and the converse is true under some conditions. Various characterizations of $S-r-$ideals are introduced. In addition, the $S-r-$ideal concept is examined under ring homomorphism, Cartesian product, amalgamation, and trivial extension. In conclusion, $S-r-$ideals are studied in polynomial rings and it is investigated that when $A[x]$ is an $S-r-$ideal of $H[x].$