Pseudo-Dedekind domains and divisorial ideals in \(R[X]_ T\)
type: Item
instance of: scholarly article
title: Pseudo-Dedekind domains and divisorial ideals in \(R[X]_ T\)
zbMATH Open document ID: 0698.13004
DOI: 10.1016/0021-8693(89)90219-6
published in: Journal of Algebra
publication date: 1989-01-01T00:00:00Z
review text: The authors introduce a notion of pseudo-Dedekind domain as a domain \(R\) in which every divisorial ideal is invertible. This generalizes the notion of pseudo-principal domain introduced by \textit{Bourbaki} and the notion of G-GCD domain (for which finite type divisorial ideals are invertible). The authors give several alternative characterizations of pseudo-Dedekind domains, for example, for an integral domain R the following conditions are equivalent: (1) \(R\) is pseudo-Dedekind, (2) \((A.B)^{-1}=A^{-1}.B^{-1}\) for all nonzero fractional ideals of \(R\), (3) \(R\) is completely integrally closed and the product of divisorial ideals is divisorial. Further, using some results which give cases when the extension or contraction of divisorial ideals remain divisorial they prove that the following statements are equivalent: (1) \(R\) is pseudo-Dedekind, (2) \(R[X]\) is pseudo-Dedekind, (3) \(R(X)\) is pseudo-Dedekind, (4) \(R(X)\) is pseudo-principal. Moreover, a group ring \(R[X,G]\) is pseudo-Dedekind if and only if \(R\) is pseudo-Dedekind and \(G\) has type \((0,0,...)\).
Mathematics Subject Classification ID: 13F05
Mathematics Subject Classification ID: 13F20
zbMATH DE Number: 4144103
zbMATH Keywords: polynomial ring
zbMATH Keywords: ring of rational functions
zbMATH Keywords: pseudo-Dedekind domain
author: D. D. Anderson
author: B. G. Kang
reviewed by: Jiří Močkoř
MaRDI profile type: Publication
full work available at URL: 0021-8693(89)90219-6
OpenAlex ID: W1978620746
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description: scientific article; zbMATH DE number 4144103