Journal article
Formally Integrally Closed Domains and the RingsR((X)) andR{{X}}
Journal of algebra, Vol.200(1), pp.347-362
02/01/1998
DOI: 10.1006/jabr.1997.7262
Abstract
LetRbe an integral domain. Forf∈R [ X ] letAfbe the ideal ofRgenerated by the coefficients off. We defineRto be formally integrally closed ⇔ (Afg)t=(AfAg)tfor all nonzerof,g∈R [ X ] . Examples of formally integrally closed domains include locally finite intersections of one-dimensional Prüfer domains (e.g., Krull domains and one-dimensional Prüfer domains). We study the ringsR((X))=R [ X ] NandR{{X}}=R [ X ] NtwhereN={f∈R [ X ] |Af=R} andNt={f∈R [ X ] |(Af)t=R}. We show thatRis a Krull domain (resp., Dedekind domain) ⇔R{{X}} (resp.,R((X))) is a Krull domain (resp., Dedekind domain) ⇔R{{X}} (resp.,R((X))) is a Euclidean domain ⇔ every (principal) ideal ofR{{X}} (resp.,R((X))) is extended fromR⇔Ris formally integrally closed and every prime ideal ofR{{X}} (resp.,R((X))) is extended fromR.
Details
- Title: Subtitle
- Formally Integrally Closed Domains and the RingsR((X)) andR{{X}}
- Creators
- D.D Anderson - Department of Mathematics, The University of Iowa, Iowa City, Iowa, 52242B.G Kang - Department of Mathematics, Pohang Institute of Science and Technology, Pohang, 790-784, Korea
- Resource Type
- Journal article
- Publication Details
- Journal of algebra, Vol.200(1), pp.347-362
- DOI
- 10.1006/jabr.1997.7262
- ISSN
- 0021-8693
- eISSN
- 1090-266X
- Publisher
- Elsevier Inc
- Language
- English
- Date published
- 02/01/1998
- Academic Unit
- Mathematics
- Record Identifier
- 9983985938102771
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