Journal article
GCD-sets in integral domains
Houston Journal of Mathematics, Vol.25(1), pp.15-34
1999
Abstract
Let R be an integral domain. A saturated multiplicative subset S ≠ U(R) of R is a GCD-set if gcd(a, b) exists for each a, b ∈ S. We study the structure of GCD-sets of R, with emphasis on the case where R is a Dedekind domain. We show that if R is atomic, then each GCD-set is generated by completely irreducible elements, and that if R is a Dedekind domain and cursive Greek chi is a nonzero nonunit of R, then for some N ≥ 1, cursive Greek chiN has a completely irreducible factor. Let R be a Dedekind domain with torsion realizable pair {Cl(R), A}. If S is a GCD-set of R, then there is a subgroup GS of Cl(R) generated by an independent subset of A with Cl(R)/GS ≅ Cl(RS). Conversely, suppose that G is a subgroup of Cl(R) generated by an independent subset of A. Then there is a GCD-set SG of R with Cl(R)/G ≅ Cl(RSG).
Details
- Title: Subtitle
- GCD-sets in integral domains
- Creators
- D.D. AndersonD.F. AndersonJ. Park
- Resource Type
- Journal article
- Publication Details
- Houston Journal of Mathematics, Vol.25(1), pp.15-34
- ISSN
- 0362-1588
- Language
- English
- Date published
- 1999
- Academic Unit
- Mathematics
- Record Identifier
- 9984230628302771
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