Pro-injective Module And Pro-flat Module

In Homologic Algebra,injective modules,projective modules and flat mod-ules play important roles,while quasi-injective is also a hot research in recently years.This paper mainly introduce pro-injective module and pro-flat module.It is consist of three parts.In the first chapter we introduce the background of this paper,some prop-erties which has been proved over the injective module and quasi-injective module.In the second chapter,first we introduce the concept of pro-injective mod-ule and some equal conditions,and then give some properties about pro-injective module.The direct summand of pro-injective is also pro-injective,and when U is generator,U is pro-injective if and only if U is injective module have been proved. In some special rings,the some character of pro-injective,injective and quasi-injective have been discussed.Because of the well duality of injective module and flat module we introduce pro-flat module and has proved that a R-module U is pro-flat module if and only if its character module U* is a pro-injective module.In the last chapter,we introduce the pro-injective resolution,the pro-flat resolution and the dimensions of them.It is proved that a module U is a pro-injective module if and only if PIExtR1(A, U)=0 for any module A.