The class group of \(D+M\)
type: Item
instance of: scholarly article
title: The class group of \(D+M\)
zbMATH Open document ID: 0668.13013
DOI: 10.1016/0022-4049(88)90091-6
author: David F. Anderson
author: Alain Ryckaert
published in: Journal of Pure and Applied Algebra
publication date: 1988-01-01T00:00:00Z
review text: Let T be an integral domain of the form \(K+M\), where K is a field and M is a (non-zero) maximal ideal. This note is concerned with the class group Cl(R), Picard group Pic(R) and local class group G(R) for a subring of T of the form \(R=D+M\), where D is a ring contained in K. When K is the quotient field of D the inclusion maps induce an exact sequence \(0\to F(D)\to F(R)\to F(T)\) for \(F=Cl, Pic\) and G; in particular when \(F=Pic\) the last homomorphism is onto and the resulting short exact sequence splits. Im(Cl(R)\(\to Cl(T))\) is identified in a number of special cases including \(Im=Pic(T)\), when this occurs \(Cl(R)=Cl(D)\oplus Pic(T)\) and \(G(R)=G(D)\). As an application of this calculation of Cl(R), a criterion for R to be a PVMD is established. The note ends by showing how the D\(+M\) construction provides examples of rings with arbitrarily preassigned class groups.
Mathematics Subject Classification ID: 13C99
Mathematics Subject Classification ID: 13B02
Mathematics Subject Classification ID: 13A05
Mathematics Subject Classification ID: 13G05
Mathematics Subject Classification ID: 14C22
Mathematics Subject Classification ID: 13F05
zbMATH DE Number: 4093612
zbMATH Keywords: Prüfer v-multiplication domain
zbMATH Keywords: integral domain
zbMATH Keywords: Picard group
zbMATH Keywords: local class group
zbMATH Keywords: \(D+M\) construction
reviewed by: David Kirby
MaRDI profile type: Publication
full work available at URL: 0022-4049(88)90091-6
OpenAlex ID: W2087518277
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description: scientific article; zbMATH DE number 4093612