Almost prime ideals in noncommutative rings

TL;DR

This paper extends the concept of almost prime ideals from commutative to noncommutative rings, providing definitions, equivalences, and exploring conditions where all right ideals are almost prime.

Contribution

It introduces the notion of almost prime ideals in noncommutative rings and explores their properties and special cases.

Findings

Defined almost prime ideals in noncommutative rings

Provided equivalent characterizations of these ideals

Identified conditions where all right ideals are almost prime

Abstract

A proper ideal $P$ of a commutative ring with identity is an almost prime ideal if $ab \in P{\setminus}P^2$ implies $a \in P$ or $b \in P$. In this paper we define almost prime ideals of a noncommutative ring, and provide some equivalent definitions. We also examine some cases such that all right ideals of a noncommutative ring are almost prime right ideals.