Generalizations of almost prime and right $S$-prime ideals in noncommutative rings

TL;DR

This paper extends the theory of almost prime and right S-prime ideals in noncommutative rings, introducing almost right S-prime ideals and exploring their properties and constructions.

Contribution

It introduces the concept of almost right S-prime ideals and analyzes their behavior in various ring structures, expanding the understanding of prime-like ideals in noncommutative rings.

Findings

Characterization of almost right S-prime ideals in different ring classes

Behavior of these ideals under ring homomorphisms and quotients

Construction of such ideals via Nagata idealization method

Abstract

Let $R$ be a noncommutative ring, and let $S$ be an $m$-system of $R$. In this paper, we give more results on the concept of almost prime (right) ideals, that were introduced by the first two authors, especially in (right) $S$-unital rings, local rings, and decomposable rings. In addition, we introduce the concept of almost right $S$-prime ideals, and we show how some findings regarding almost prime ideals can be derived as consequences of almost right $S$-prime ideals. Besides, we show how almost right $S$-prime ideals behave in related rings such as homomorphic images, quotient rings, and decomposable rings. Finally, we construct almost right $S$-prime ideals using the Nagata method of idealization.