Journal article
Condensed domains
Canadian Mathematical Bulletin, Vol.46(1), pp.3-13
2003
DOI: 10.4153/CMB-2003-001-2
Abstract
Abstract An integral domain D with identity is condensed (resp., strongly condensed) if for each pair of ideals I, J of D, IJ = { ij ; i ∈ I; j ∈ J } (resp., IJ = iJ for some i ∈ I or IJ = Ij for some j ∈ J ). We show that for a Noetherian domain D, D is condensed if and only if Pic( D ) = 0 and D is locally condensed, while a local domain is strongly condensed if and only if it has the two-generator property. An integrally closed domain D is strongly condensed if and only if D is a Bézout generalized Dedekind domain with at most one maximal ideal of height greater than one. We give a number of equivalencies for a local domain with finite integral closure to be strongly condensed. Finally, we show that for a field extension k ⊆ K , the domain D = k + XK [[ X ]] is condensed if and only if [ K : k ] ≤ 2 or [ K : k ] = 3 and each degree-two polynomial in k [ X ] splits over k , while D is strongly condensed if and only if [ K : k ] ≤ 2.
Details
- Title: Subtitle
- Condensed domains
- Creators
- D D Anderson - University of Iowa, MathematicsTiberiu Dumitrescu
- Resource Type
- Journal article
- Publication Details
- Canadian Mathematical Bulletin, Vol.46(1), pp.3-13
- DOI
- 10.4153/CMB-2003-001-2
- ISSN
- 0008-4395
- eISSN
- 1496-4287
- Language
- English
- Date published
- 2003
- Academic Unit
- Mathematics
- Record Identifier
- 9983985706502771
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