The splitting of finitely generated modules
type: Item
instance of: scholarly article
title: The splitting of finitely generated modules
zbMATH Open document ID: 0598.13003
author: Jagannadham V. Pakala
author: Thomas S. Shores
published in: Osaka Journal of Mathematics
publication date: 1985-01-01T00:00:00Z
review text: The authors mention that the purpose of this paper is to study splitting properties of modules with respect to their torsion submodule, with special attention to the cases of integral domains and (von Neumann) regular rings. A torsion theory is said to have the finitely generated splitting property (FGSP) if the torsion submodule of every finitely generated module splits off as a direct summand of the module. A weaker condition is the cyclic splitting property (CSP): the torsion submodule of every cyclic module splits off. A classical result due to Kaplansky is that the usual torsion theory for an integral domain R has FGSP if and only if R is a Prüfer domain. The authors extend this result to arbitrary torsion theories over an integral domain, obtaining Kaplansky's result as a special case. The CSP property is related to the vanishing of certain Tor-groups, and also, especially for regular rings, to more intrinsic conditions. The authors also answer a question of \textit{K. Oshiro} [Osaka J. Math. 15, 371-380 (1978; Zbl 0396.13007)] concerning FGSP for regular rings.
Mathematics Subject Classification ID: 13C05
Mathematics Subject Classification ID: 13D30
Mathematics Subject Classification ID: 13E15
zbMATH DE Number: 3963992
zbMATH Keywords: von Neumann regular rings
zbMATH Keywords: integral domains
zbMATH Keywords: torsion theory
zbMATH Keywords: finitely generated splitting property
zbMATH Keywords: cyclic splitting property
zbMATH Keywords: FGSP
zbMATH Keywords: CSP
reviewed by: William J. Heinzer
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3963992