On phi-1-absorbing prime ideals

Yıldız E., Tekir Ü., Koç S.

BEITRAGE ZUR ALGEBRA UND GEOMETRIE-CONTRIBUTIONS TO ALGEBRA AND GEOMETRY, vol.62, no.4, pp.907-918, 2021 (Peer-Reviewed Journal) identifier identifier

  • Publication Type: Article / Article
  • Volume: 62 Issue: 4
  • Publication Date: 2021
  • Doi Number: 10.1007/s13366-020-00557-w
  • Journal Name: BEITRAGE ZUR ALGEBRA UND GEOMETRIE-CONTRIBUTIONS TO ALGEBRA AND GEOMETRY
  • Journal Indexes: Emerging Sources Citation Index, Scopus, MathSciNet, zbMATH
  • Page Numbers: pp.907-918
  • Keywords: phi-prime ideal, 1-absorbing prime ideal, phi-1-absorbing prime ideal, Generalizations of prime ideal, 2-ABSORBING IDEALS

Abstract

In this paper, we introduce phi-1-absorbing prime ideals in commutative rings. Let R be a commutative ring with a nonzero identity 1 not equal 0 and phi : I(R) -> I(R) boolean OR {theta} be a function where I( R) is the set of all ideals of R. A proper ideal I of R is called a phi-1-absorbing prime ideal if for each nonunits x, y, z is an element of R with xyz is an element of I - phi(I), then either xy is an element of I or z is an element of I. In addition to give many properties and characterizations of phi-1-absorbing prime ideals, we also determine rings in which every proper ideal is phi-1-absorbing prime.