Generalization of $z$-ideals in right duo rings
Autor: | Maryam Masoudi Arani, Reza Jahani-Nezhad |
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Rok vydání: | 2020 |
Předmět: |
Statistics and Probability
Matematik Ring (mathematics) Algebra and Number Theory z-ideal Duo ring Relative z-ideal Semisimple ring von Neumann regular ring Mathematics::Commutative Algebra Generalization Mathematics::Rings and Algebras Minimal prime ideal Structure (category theory) Noncommutative geometry Combinatorics Intersection Von Neumann regular ring Geometry and Topology Ideal (ring theory) Mathematics Analysis |
Zdroj: | Volume: 49, Issue: 4 1423-1436 Hacettepe Journal of Mathematics and Statistics |
ISSN: | 2651-477X |
Popis: | 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. |
Databáze: | OpenAIRE |
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