Journal article
A Class of Atomic Rings
Communications in Algebra, Vol.40(3), pp.1086-1095
03/01/2012
DOI: 10.1080/00927872.2010.544698
Abstract
Let R be a commutative ring with identity. A nonunit element a ∈ R is an atom if a = bc (b, c ∈ R) implies (a) = (b) or (a) = (c), and R is atomic if each nonunit of R is a finite product of atoms. A ring R is a (weak) CK ring if R is atomic and (each maximal ideal of) R contains only finitely many nonassociate atoms. We show that the following are equivalent: (1) R is a weak CK ring, (2) each (prime) ideal of R is a finite union of principal ideals, (3) R is atomic and each maximal ideal is a finite union of principal ideals, and (4) R is a finite direct product of finite local rings, special principal ideal rings (SPIRs), and (one-dimensional) Noetherian domains in which every maximal ideal is a finite union of principal ideals (or equivalently, weak CK domains). We show that a domain R is a weak CK domain if and only if R is a Noetherian domain with R M a CK domain for each maximal ideal M of R and Pic(R) = 0.
Details
- Title: Subtitle
- A Class of Atomic Rings
- Creators
- Sangmin Chun - Department of Mathematics , Seoul National UniversityD. D Anderson - Department of Mathematics , The University of Iowa
- Resource Type
- Journal article
- Publication Details
- Communications in Algebra, Vol.40(3), pp.1086-1095
- Publisher
- Taylor & Francis Group
- DOI
- 10.1080/00927872.2010.544698
- ISSN
- 0092-7872
- eISSN
- 1532-4125
- Language
- English
- Date published
- 03/01/2012
- Academic Unit
- Mathematics
- Record Identifier
- 9983985875302771
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