Purity And Projectivity Relative To A Submodule

In the literature there are some links between the notions ofdirectsummands, purity and projectivity. In this paper, we study theselinks among the notions but relative to an arbitrary submodules, also apartial links are consider. Properties and characterizations of thesenotions which are correspondence, have been given.Among others, thefollowing have been proved. If M is a projective module, every puresubmoduleN relative to a submodule T is projective relative to N