Notes on 1-Absorbing Prime Ideals

Authors

  • El Mehdi Bouba Team of Modeling and Scientific Computing, Mathematics Department, Pluridisciplinary Faculty, Mohammed First University
  • Mohammed Tamekkante Department of Mathematics, Faculty of Science, University Moulay Ismail Meknes
  • Ünsal Tekir Department of Mathematics, Marmara University, Istanbul
  • Suat Koç Department of Mathematics, Istanbul Medeniyet University

DOI:

https://doi.org/10.7546/CRABS.2022.05.01

Keywords:

primary ideal, 1-absorbing primary ideal, 1-absorbing prime ideal

Abstract

Let R be a commutative ring with a nonzero identity. A proper ideal I of R is said to be a 1-absorbing prime ideal if xyzI for some nonunits x, y, zR, then xyI or zI. It is well known that prime ideal ⇒ 1-absorbing prime idealprimary ideal ⇒ semi-primary ideal, that is, the class of 1-absorbing prime ideals comes between the classes of prime ideals and primary ideals. Also, the above right arrows are not reversible. In this article, we characterize rings over which every 1-absorbing prime ideal is prime and every primary ideal is 1-absorbing prime. Also, by comparing 1-absorbing prime ideals and other some classical ideals such as 2-absorbing ideals and semi-primary ideals, we characterize Noetherian divided rings and von Neumann regular rings. 

Author Biographies

El Mehdi Bouba, Team of Modeling and Scientific Computing, Mathematics Department, Pluridisciplinary Faculty, Mohammed First University

Mailing Address:
Team of Modeling and Scientific Computing,
Mathematics Department,
Pluridisciplinary Faculty,
Mohammed First University
B.P. 300, Selouane, Nador 62700, Morocco

E-mail: mehdi8bouba@hotmail.fr

Mohammed Tamekkante, Department of Mathematics, Faculty of Science, University Moulay Ismail Meknes

Mailing Address:
Department of Mathematics,
Faculty of Science,
University Moulay Ismail Meknes
Box 11201, Zitoune, Morocco

E-mail: tamekkante@yahoo.fr

Ünsal Tekir, Department of Mathematics, Marmara University, Istanbul

Mailing Address:
Department of Mathematics,
Marmara University
Istanbul, Turkey

E-mail: utekir@marmara.edu.tr

Suat Koç, Department of Mathematics, Istanbul Medeniyet University

Mailing Address:
Department of Mathematics,
Istanbul Medeniyet University
Istanbul, Turkey

E-mail: suat.koc@medeniyet.edu.tr

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Published

31-05-2022

How to Cite

[1]
E. M. Bouba, M. Tamekkante, Ünsal Tekir, and S. Koç, “Notes on 1-Absorbing Prime Ideals”, C. R. Acad. Bulg. Sci. , vol. 75, no. 5, pp. 631–639, May 2022.

Issue

Section

Mathematics