Finitely generated projective modules over exchange rings
type: Item
instance of: scholarly article
title: Finitely generated projective modules over exchange rings
zbMATH Open document ID: 0824.16002
DOI: 10.1007/BF02567984
author: Tong Wenting
author: Tongsuo Wu
published in: Manuscripta Mathematica
publication date: 1995-04-03T00:00:00Z
full work available at URL: https://eudml.org/doc/156044
review text: The authors study finitely generated projective modules over exchange rings. They prove that a finitely generated projective module over an exchange ring \(R\) is isomorphic to a direct sum of cyclic \(R\)-modules generated by idempotents. If \(R\) is an exchange ring such that \(R/J(R)\) is a ring with central idempotents, then the cancellation law holds in \(p(R)\), the category of all finitely generated projective \(R\)-modules. If \(R\) is an exchange ring with central idempotents then the family of all nonnegative integral-valued continuous functions on \(\text{Spec} (R)\) is precisely the family of all rank functions from \(\text{Spec} (R)\) into \(Z_ +\) induced by finitely generated projective \(R\)-modules. Finally, if \(R\) is an exchange ring with central idempotents, there is an order- preserving isomorphism between \((K_ 0 (R), [R])\) and \((H_ 0 (R), u)\), where \(K_ 0(R)\) is the Grothendieck group of \(R, H_ 0(R)\) is the set of all continuous functions from \(\text{Spec} (R)\) into \(Z\) and \(u\) is the order unit of \(H_ 0 (R)\).
Mathematics Subject Classification ID: 16D40
Mathematics Subject Classification ID: 16E20
Mathematics Subject Classification ID: 16S60
zbMATH DE Number: 733197
zbMATH Keywords: spectra of rings
zbMATH Keywords: finitely generated projective modules
zbMATH Keywords: exchange rings
zbMATH Keywords: direct sum of cyclic modules
zbMATH Keywords: central idempotents
zbMATH Keywords: cancellation
zbMATH Keywords: rank functions
zbMATH Keywords: Grothendieck groups
reviewed by: Veereshwar A. Hiremath
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 733197