Some Properties Of Projective Cover And Injective Envelope

Projective modules and injective modules are the most fundamental mod-ules,related to these are the concepts of projective covers and injective en-velopes.This passage gives some properties of projective cover and injective envelope systematically.Firstly,This passage introduces some properties of pro-jective modules and injective modules,discusses the direct sum and direct prod-uct of projective modules and injective modules.In Chapter 4 we give three equivalent definitions of projective cover and injective envelope.For arbitrary R-module,the injective cover always exits,but the projective cover doesn’t al-ways exist.For example the finitely generated modules over integral ring don’t have projective covers.We give a theorem about when the projective cover doesn’t exist.We also give some properties of projective cover.For example,A is the superfluous submodule of M and M has a projective cover if and only if M/A has a projective cover. In Chapter 5 we discuss the properties of mod-ules over perfect ring and give the dual theorems about projective cover and injective envelope.Finally,we discuss the concepts and uniqueness of minimal projective resolution and minimal injective resolution.