Purity relative to a cyclic module

Abstract

We study relative projectivity and injectivity classes of exact sequences with respect to the classes of cyclic modules. A characterization of cyclic pure exact sequences has given in terms of exactness of a certain sequence of submodules of the modules appearing in the given sequence. We also study the concept of relative divisibility of elements in submodules (known as - purity). We give the generalization of the proposition of Stenstrom. We characterize the preservation of exactness by cyclic modules where be a left ideal of the ring . We also relate the purity in Quasi- projective module [9]. We also try to define Copure for a class of modules and co-relate Copure injective or Copure projective with it. We derive some results dualize certain results of R.B. Warfield.