ABSOLUTELY F/U-PURE MODULES

Abstract

 Let R be an associative ring with non-zero identity. A submodule A of a right R-module B is said to be F/U-pure if fÄ_R 1_F/U is a monomorphism for every free left R-module F and for every cyclic sub-module U of F, where f:A®B is the inclusion monomorphism. A right R-module D is said to be absolutely F/U-pure if D is F/U-pure in every right R-module which contains it as a submodule. We characterize absolutely F/U-purity by injectivity with respect to a certain monomorphism. We also prove that the class of absolutely F/U-pure right R-modules is closed under taking direct products, direct sums and extensions. Moreover, we consider absolutely F/U-pure right modules over right noetherian rings and regular (von Neumann) rings.