AI Summary & Key Takeaways
AI InsightMoreover, 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.
📌 3 Key Takeaways
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.