Some Studies On Quasi EP-injective Modules And Their Endomorphism Rings

In this paper,we introduce the conception of quasi EP-injective modules.The properties of quasi EP-injective modules are gived.and we characterise some special rings by using quasi EP-injective modules.In the same time,we study the endomorphism rings of quasi EP-injective modules.This paper contains two parts.In the first part,we generalize the conception and basic properties of quasi GP-injective modules into quasi EP-injective modules,obtain the following results:Let M be a right R-module,S=End(MR).(1)Suppose S is a right EP-injective ring,then M is a right quasi EP-injective mod-ule.(2)If M is a right quasi EP-injective module,and M generates ker s for all s which is in S.(3)Suppose M is a self-generated quasi EP-injective module,S is a right duo ring.then J(S)(?)Δ.In the second part,we mainly give the semisimplicity,regularity of endomorphism rings of quasi EP-injective modules.The relationship-s between some special rings and the endomorphism rings of quasi EP-injective modules are studied.It is proved that:Let Mr be a quasi EP-injective module,S=End (MR) satisfises the special ascending chain condition.(1)If S is a J-ring,then S/J(S) is a semisimple ring,specially,it is a strongly regular ring.(2)If S is an exchange ring.then the follow-ing conditions are equivalent:(a)S/J(S) is a strongly regular ring;(b)S is a N-ring;(c)S/J(S) is a reduced ring;(d)S/J(S) is a semiabelian ring;(e)S/J(S) is an abelian ring.