The splitting of finitely generated modules
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.
scientific article; zbMATH DE number 3963992
scientific article