Year 2020, Volume 49 , Issue 4, Pages 1423 - 1436 2020-08-06

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
  • [1] A.R. Aliabad, F. Azarpanah and A. Taherifar, Relative z-ideals in commutative rings, Comm. Algebra, 41, 325–341, 2013.
  • [2] F. Azarpanah and A. Taherifar, Relative z-ideals in C(X), Topology Appl. 156, 1711–1717, 2009
  • [3] R.C. Courter, Finite dimensional right duo algebras are duo, Proceedings of the Amer. Math. Soc. 84 (2), 157–161, 1982.
  • [4] L. Gillman and M. Jerison, Rings of continuous functions, The University Series in Higher Mathematics, New York, Van Nostrand, 1960.
  • [5] N.K. Kim and Y. Lee, On a ring property unifying reversible and right duo rings, J. Korean Math. Soc. 50 (5), 1083-1103, 2013.
  • [6] C.W. Kohls, Ideals in rings of continuous functions, Fund. Math. 45, 28–50, 1957.
  • [7] T.Y. Lam, A first course in noncommutative ring, Graduate Texts in Mathematics 131, Springer-Verlag, New York, 1991.
  • [8] T.Y. Lam and A.S. Dugas, Quasi-duo rings and stable range descent, J. Pure Appl. Algebra 195, 243–259, 2005.
  • [9] G. Marks, Duo rings and ore extensions, J. Algebra, 280, 463–471, 2004.
  • [10] G. Mason, $z$-ideals and prime ideals, J. Algebra, 26, 280–297, 1973.
  • [11] G. Mason, Prime $z$-ideals of $C(X)$ and related rings, Canad. Math. Bull. 23 (4), 437–443, 1980.
  • [12] A. Rezaei Aliabad and R. Mohamadian, On $z$-ideals and $z^\circ$-ideals of Power Series Rings, J. Math. Ext. 7 (2), 93–108, 2013.
Primary Language en
Subjects Mathematics
Journal Section Mathematics
Authors

Orcid: 0000-0002-5165-486X
Author: Maryam MASOUDİ-ARANİ
Institution: University of Kashan
Country: Iran


Orcid: 0000-0001-8207-1969
Author: Reza JAHANİ-NEZHAD (Primary Author)
Institution: Technical and Vocational University
Country: Iran


Dates

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 <https://dergipark.org.tr/en/pub/hujms/issue/56305/536025>
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.