Structural Analysis of Commutative S-Reduced Rings

TL;DR

This paper investigates the structure of $S$-reduced rings, establishing key properties, relationships with other classes of rings, and providing a structure theorem for these rings.

Contribution

It introduces new structural results for $S$-reduced rings, including their intersection properties, isomorphisms, and relationships with $S$-Armendariz and $S$-strongly Hopfian rings.

Findings

The intersection of all $S$-prime ideals in an $S$-reduced ring is $S$-zero.

An $S$-Artinian reduced ring is isomorphic to a finite product of fields.

The class of $u$-$S$-reduced rings is contained within $u$-$S$-Armendariz$ rings.

Abstract

Let $R$ be a commutative ring with identity, $S \subseteq R$ be a multiplicative set. In this paper, we establish that the intersection of all $S$-prime ideals in an $S$-reduced ring is $S$-zero. Also, we show that an $S$-Artinian reduced ring is isomorphic to the finite direct product of fields. Furthermore, we provide an example of an $S$-reduced ring which is a uniformly-$S$-Armendariz ring (in short, $u$-$S$-Armendariz$)$ ring. Additionally, we prove that the class of uniformly-$S$-reduced rings (in short, $u$-$S$-reduced rings) belongs to the class of $u$-$S$-Armendariz rings. Among other results, we establish the relationship between $S$-reduced rings and $S$-strongly Hopfian rings. Finally, we prove the structure theorem for $S$-reduced rings.