Almost prime ideals in noncommutative rings

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.