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## Generalization of $z$-ideals in right duo rings

#### Maryam MASOUDİ-ARANİ [1] , Reza JAHANİ-NEZHAD [2]

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.
z-ideal, Duo ring, Relative z-ideal, Semisimple ring, von Neumann regular ring
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Primary Language en Mathematics Mathematics Orcid: 0000-0002-5165-486XAuthor: Maryam MASOUDİ-ARANİ Institution: University of KashanCountry: Iran Orcid: 0000-0001-8207-1969Author: Reza JAHANİ-NEZHAD (Primary Author)Institution: Technical and Vocational UniversityCountry: Iran Publication Date : August 6, 2020
 Bibtex @research article { hujms536025, journal = {Hacettepe Journal of Mathematics and Statistics}, issn = {2651-477X}, eissn = {2651-477X}, address = {}, publisher = {Hacettepe University}, year = {2020}, volume = {49}, pages = {1423 - 1436}, doi = {10.15672/hujms.536025}, title = {Generalization of \$z\$-ideals in right duo rings}, key = {cite}, author = {Masoudi̇-arani̇, Maryam and Jahani̇-nezhad, Reza} } APA Masoudi̇-arani̇, M , Jahani̇-nezhad, R . (2020). Generalization of $z$-ideals in right duo rings . Hacettepe Journal of Mathematics and Statistics , 49 (4) , 1423-1436 . DOI: 10.15672/hujms.536025 MLA Masoudi̇-arani̇, M , Jahani̇-nezhad, R . "Generalization of $z$-ideals in right duo rings" . Hacettepe Journal of Mathematics and Statistics 49 (2020 ): 1423-1436 Chicago Masoudi̇-arani̇, M , Jahani̇-nezhad, R . "Generalization of $z$-ideals in right duo rings". Hacettepe Journal of Mathematics and Statistics 49 (2020 ): 1423-1436 RIS TY - JOUR T1 - Generalization of $z$-ideals in right duo rings AU - Maryam Masoudi̇-arani̇ , Reza Jahani̇-nezhad Y1 - 2020 PY - 2020 N1 - doi: 10.15672/hujms.536025 DO - 10.15672/hujms.536025 T2 - Hacettepe Journal of Mathematics and Statistics JF - Journal JO - JOR SP - 1423 EP - 1436 VL - 49 IS - 4 SN - 2651-477X-2651-477X M3 - doi: 10.15672/hujms.536025 UR - https://doi.org/10.15672/hujms.536025 Y2 - 2019 ER - EndNote %0 Hacettepe Journal of Mathematics and Statistics Generalization of $z$-ideals in right duo rings %A Maryam Masoudi̇-arani̇ , Reza Jahani̇-nezhad %T Generalization of $z$-ideals in right duo rings %D 2020 %J Hacettepe Journal of Mathematics and Statistics %P 2651-477X-2651-477X %V 49 %N 4 %R doi: 10.15672/hujms.536025 %U 10.15672/hujms.536025 ISNAD Masoudi̇-arani̇, Maryam , Jahani̇-nezhad, Reza . "Generalization of $z$-ideals in right duo rings". Hacettepe Journal of Mathematics and Statistics 49 / 4 (August 2020): 1423-1436 . https://doi.org/10.15672/hujms.536025 AMA Masoudi̇-arani̇ M , Jahani̇-nezhad R . Generalization of $z$-ideals in right duo rings. Hacettepe Journal of Mathematics and Statistics. 2020; 49(4): 1423-1436. Vancouver Masoudi̇-arani̇ M , Jahani̇-nezhad R . Generalization of $z$-ideals in right duo rings. Hacettepe Journal of Mathematics and Statistics. 2020; 49(4): 1423-1436.

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