The Preservation Of Covers And Envelopes Of Modules

In this thesis,we talk about the preservation of covers and envelopes of modules.In section two,we focus on the following two problems:Under what conditions,the direct product of injective envelopes of modules is an injective envelope of the direct product of modules?Under what conditions,the direct sum of projective covers of modules is a projective cover of the direct sum of modules?The first question has been answered in some articles([5],[10]):?E(M_i)? E(?M_i)if and only if R is semi-artinian.We will give some equivalent characterizations for the second problem and prove:when one of these conditions is satisfied,?P(M_i)=P(?M_i).In the third section of this thesis,we focus on general covers,envelopes and general adjoint functors and consider:Given an adjoint pair(F.G),under what conditions,F would preserve general envelopes?And under what conditions,G would preserve general covers?We will give our answers to these questions.