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

TL;DR

This paper explores the properties, characterizations, and extensions of S-r-ideals in commutative rings, including their behavior in polynomial rings and under various ring constructions.

Contribution

It introduces the concept of S-uz-rings, characterizes S-r-ideals in different contexts, and examines their behavior in polynomial rings and ring extensions.

Findings

Characterization of S-r-ideals in commutative rings

Definition and characterization of S-uz-rings

Behavior of S-r-ideals in polynomial rings

Abstract

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 $H[x].$