Journal article
Commutative rings with finitely generated monoids of fractional ideals
Journal of algebra, Vol.320(7), pp.3006-3021
2008
DOI: 10.1016/j.jalgebra.2008.06.032
Abstract
Let R be a commutative ring with identity and let P ( R ) be the monoid of principal fractional ideals of R. We show that P ( R ) is finitely generated if and only if P ( R ¯ ) ( R ¯ the integral closure of R) is finitely generated and R ¯ / [ R : R ¯ ] is finite. Moreover, R ¯ is a finite direct product of finite local rings, SPIRs, Bezout domains D with P ( D ) finitely generated, and special Bezout rings S ( S is a Bezout ring with a unique minimal prime P, S P is an SPIR, and P ( S / P ) is finitely generated). Also, P ( R ) is finitely generated if and only if F * ( R ) , the monoid of finitely generated fractional ideals of R, is finitely generated. We show that the monoid F ( R ) of fractional ideals of R is finitely generated if and only if the monoid F ¯ ( R ) of R-submodules of the total quotient ring of R is finitely generated and characterize the rings for which this is the case.
Details
- Title: Subtitle
- Commutative rings with finitely generated monoids of fractional ideals
- Creators
- D.D AndersonS Chun
- Resource Type
- Journal article
- Publication Details
- Journal of algebra, Vol.320(7), pp.3006-3021
- DOI
- 10.1016/j.jalgebra.2008.06.032
- ISSN
- 0021-8693
- eISSN
- 1090-266X
- Publisher
- Elsevier Inc
- Language
- English
- Date published
- 2008
- Academic Unit
- Mathematics
- Record Identifier
- 9983985826302771
Metrics
19 Record Views