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 2 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 2 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 ideal of a p-right duo ring R is a left relative z-ideal, then every ideal of R is a z-ideal.