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Divisorial domains

Divisorial domains Abstract. Let R be a domain with quotient ®eld Q. R is divisorial if R X R X I I for every nonzero fractional ideal I of R. We prove that a local domain R, not a ®eld, is divisorial if and only if QaR has simple essential socle and RarR is AB-5* for every nonzero r e R. We give examples of non-divisorial and of non-®nitely divisorial local domains such that QaR has simple essential socle. If A is any R-submodule of Q with endomorphism ring R, we say that R is A-divisorial if A X A X X X for every nonzero submodule X of A. We prove that if a local noetherian domain R is A-divisorial for some A, then R is one-dimensional and A is ®nitely generated, i.e. A is isomorphic to a canonical ideal of R. If A is a fractional ideal of R we generalize the characterization of divisorial domains, namely we prove that R is Adivisorial if and only if QaA has simple essential socle and RarR is AB-5* for every nonzero r e R. 1991 Mathematics Subject Classi®cation: 13G05, 13C13; 13A15, 13B22. Introduction Throughout this paper, R will denote an http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Forum Mathematicum de Gruyter

Divisorial domains

Bazzoni, Silvana
Forum Mathematicum , Volume 12 (4) – May 29, 2000
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Publisher
de Gruyter
Copyright
Copyright © 2000 by Walter de Gruyter GmbH & Co. KG
ISSN
0933-7741
eISSN
1435-5337
DOI
10.1515/form.2000.011
Publisher site
See Article on Publisher Site

Abstract

Abstract. Let R be a domain with quotient ®eld Q. R is divisorial if R X R X I I for every nonzero fractional ideal I of R. We prove that a local domain R, not a ®eld, is divisorial if and only if QaR has simple essential socle and RarR is AB-5* for every nonzero r e R. We give examples of non-divisorial and of non-®nitely divisorial local domains such that QaR has simple essential socle. If A is any R-submodule of Q with endomorphism ring R, we say that R is A-divisorial if A X A X X X for every nonzero submodule X of A. We prove that if a local noetherian domain R is A-divisorial for some A, then R is one-dimensional and A is ®nitely generated, i.e. A is isomorphic to a canonical ideal of R. If A is a fractional ideal of R we generalize the characterization of divisorial domains, namely we prove that R is Adivisorial if and only if QaA has simple essential socle and RarR is AB-5* for every nonzero r e R. 1991 Mathematics Subject Classi®cation: 13G05, 13C13; 13A15, 13B22. Introduction Throughout this paper, R will denote an

Journal

Forum Mathematicum – de Gruyter

Published: May 29, 2000

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