Generalization of z-ideals in right duo rings

Abstract

The aim of this paper is to generalize the notion of z-ideals to arbitrary noncommutative rings. A left (right) ideal I of a ring R is called a left (right) z-ideal if Ma ⊆ I, for each a ∈ I, where Ma is the intersection of all maximal ideals containing a. For every two left ideals I and J of a ring R, we call I a left zJ -ideal if Ma ∩ J ⊆ I, for every a ∈ I, whenever J * I and I is a zJ -ideal, we say that I is a left relative z-ideal. We characterize the structure of them in right duo rings. It is proved that a duo ring R is von Neumann regular ring if and only if every ideal of R is a z-ideal. Also, every one sided ideal of a semisimple right duo ring is a z-ideal. We have shown that if I is a left zJ -ideal of a p-right duo ring, then every minimal prime ideal of I is a left zJ -ideal. Moreover, if every proper id