New Results On $S-r-$ideals in Commutative Rings

This article studies the notion of $S-r-$ideals in commutative ring H, where S is a multiplicatively closed subset of H. Some basic properties of $S-r-$ideals are given. Various characterizations of $S-r-$ideals are presented. Also, $S-uz-$ring is defined and it is proved that H is an $S-uz-$ring if and only if every maximal ideal disjoint from S is an $S-r-$ideal provided S is finite. In addition, the $S-r-$ideal concept is examined in amalgamation and trivial extension. Finally, $S-r-$ideals are studied in polynomial rings and it is investigated that when $A[x]$ is an $S-r-$ideal of $H[x].$