Generalization of $z$-ideals in right duo rings

Year 2020, Volume: 49 Issue: 4, 1423 - 1436, 06.08.2020

Maryam Masoudi-arani Reza Jahani-nezhad

https://doi.org/10.15672/hujms.536025

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 $M_a \subseteq I$, for each $a\in I$, where $M_a$ 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 $z_J$-ideal if $M_a \cap J \subseteq I$, for every $a\in I$, whenever $ J \nsubseteq I$ and $I$ is a $z_J$-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 $z_J$-ideal of a $p$-right duo ring, then every minimal prime ideal of $I$ is a left $z_J$-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.

Keywords

z-ideal, Duo ring, Relative z-ideal, Semisimple ring, von Neumann regular ring

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