Journal article
Generalizations of Prime Ideals
Communications in Algebra, Vol.36(2), pp.686-696
01/25/2008
DOI: 10.1080/00927870701724177
Abstract
Let R be a commutative ring with identity. Various generalizations of prime ideals have been studied. For example, a proper ideal I of R is weakly prime (resp., almost prime) if a, b ∈ R with ab ∈ I − {0} (resp., ab ∈ I − I 2 ) implies a ∈ I or b ∈ I. Let φ:ℐ(R) → ℐ(R) ∪ {∅} be a function where ℐ(R) is the set of ideals of R. We call a proper ideal I of R a φ-prime ideal if a, b ∈ R with ab ∈ I − φ(I) implies a ∈ I or b ∈ I. So taking φ ∅ (J) = ∅ (resp., φ 0 (J) = 0, φ 2 (J) = J 2 ), a φ ∅ -prime ideal (resp., φ 0 -prime ideal, φ 2 -prime ideal) is a prime ideal (resp., weakly prime ideal, almost prime ideal). We show that φ-prime ideals enjoy analogs of many of the properties of prime ideals.
Details
- Title: Subtitle
- Generalizations of Prime Ideals
- Creators
- D. D Anderson - Department of Mathematics , The University of IowaMalik Bataineh - Department of Mathematics , The University of Iowa
- Resource Type
- Journal article
- Publication Details
- Communications in Algebra, Vol.36(2), pp.686-696
- Publisher
- Taylor & Francis Group
- DOI
- 10.1080/00927870701724177
- ISSN
- 0092-7872
- eISSN
- 1532-4125
- Language
- English
- Date published
- 01/25/2008
- Academic Unit
- Mathematics
- Record Identifier
- 9983985947002771
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