NOTES ON 1-ABSORBING PRIME IDEALS

Bouba E. M., Tamekkante M., Tekir Ü., Koç S.

COMPTES RENDUS DE L ACADEMIE BULGARE DES SCIENCES, vol.75, no.5, pp.631-639, 2022 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 75 Issue: 5
  • Publication Date: 2022
  • Doi Number: 10.7546/crabs.2022.05.01
  • Journal Name: COMPTES RENDUS DE L ACADEMIE BULGARE DES SCIENCES
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Aquatic Science & Fisheries Abstracts (ASFA), BIOSIS, CAB Abstracts, zbMATH
  • Page Numbers: pp.631-639
  • Keywords: primary ideal, 1-absorbing prime ideal, 1-absorbing primary ideal, RINGS
  • Marmara University Affiliated: Yes

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 xyz is an element of I for some nonunits x, y, z is an element of R, then xy is an element of I or z is an element of I. It is well known that prime ideal double right arrow 1-absorbing prime ideal double right arrow primary ideal double right arrow 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.