Strongly torsion free, copure flat and Matlis reflexive modules

Abstract

In this paper we show that if R is a ring of Krull dimension d and M is a strongly copure flat (respectively, strongly copure injective) module, then E⊗ R M (respectively, Hom R ( E, M)) is strongly cotorsion (respectively, strongly torsion free) for any injective module E. We obtain a new characterization for copure flat modules. We prove that M is copure flat if and only if Ext R 1( M, F)=0 for all flat cotorsion modules F. Moreover, if ( R, m ) is a Cohen–Macaulay local ring and M is strongly copure flat, then Hom R ( M, F) is strongly torsion free. Let ( R, m ) be a Cohen–Macaulay local ring of Krull dimension d and M be a Matlis reflexive strongly torsion free R-module. We show that M ̂ is a maximal Cohen–Macaulay R ̂ -module. Also, if R is as above and M a Matlis reflexive strongly copure injective R-module, then Hom R(E(R/ m),M) is either zero or a maximal Cohen–Macaulay R ̂ -module.