S-J-Ideals: A Study in Commutative and Noncommutative Rings

TL;DR

This paper introduces S-J-ideals in both commutative and noncommutative rings, exploring their properties, relationships, and behavior under various ring constructions, extending the concept of J-ideals.

Contribution

It defines S-J-ideals in commutative and noncommutative rings, analyzing their properties and connections to other ideal types across multiple ring constructions.

Findings

S-J-ideals generalize J-ideals in various ring contexts

In commutative rings, S-J-ideals share many properties with J-ideals

Examples illustrate the relationship between right S-prime and J-ideals

Abstract

In this paper, we introduce the concept of S-J-ideals in both commutative and noncommutative rings. For a commutative ring R and a multiplicatively closed subset S, we show that many properties of J-ideals apply to S-J-ideals and examine their characteristics in various ring constructions, such as homomorphic image rings, quotient rings, cartesian product rings, polynomial rings, power series rings, idealization rings, and amalgamation rings. In noncommutative rings, where S is an m-system, we define right S-J-ideals. We demonstrate the equivalence of S-J-ideals and right S-J-ideals in commutative rings with identity and provide examples to illustrate the connections between right S-prime ideals and J-ideals.