Generalized Hill Lemma, Kaplansky Theorem for Cotorsion Pairs And Some Applications

Authors:
Open Access
Publisher website
Google Scholar
Abstract
We generalize Hill’s lemma,in order to obtain a large family of C- filtered submodules,from a single C-filtration of a module. We use this to prove the following generalization of Kaplansky’s structure theorem for projective modules: for any ring R, a cotorsion pair (A,B) in Mod-R is of countable type if and only if every module M 2 A is A,!-filtered. We also prove rank versions of these results for torsion-free modules over commutative domains. As an application, we solve a problem of Bazzoni and Salce [3] by showing that strongly flat modules over any valuation domain coincide with the exten- sions of free modules,by divisible torsion-free modules. Another application yields a short proof of the structure of Matlis localizations of commutative rings.

This publication has 11 references indexed in Scilit: