Some Studies On GP-injective And Wnil-injective

In this paper, we mainly study some properties of these rings whoseevery simple (singular) left module is GP-injective or Wnil-injective.This paper is composed of three chapters.In the first chapter,we mainly introduce some background knowl-edge of the injectivity at home and abroad.In the second chapter, we obtain some conditions for a GP-V-ringor a GP-V′-ring to be a strongly regularity ring by using some propertiesof GW-ideals and quasi ZI-rings, and mainly prove that: let R be aring, the following conditions are equivalent:(1)R is strongly regular;(2)R is a abelian, left GP-V′-ring,whose maximal essential left idealsare GW-ideals;(3)R is a quasi ZI, left GP-V′-ring,whose maximalessential right ideals are GW-ideals.In the third chapter, we mainly give some properties and char-acterizations of Wnil-injective modules, and under the condition thatevery simple singular left module is Wnil-injective we diucuss the rela-tionship of some special rings (such as reduced rings, semiprime ringsand quasi ZI-rings).We mainly obtain some conclusions as follows:(1)Let M be a left R-module, the following conditions are equivalent:(i)Mis a left Wnil-injective module;(ii)for any0=a∈N (R), there existsa positive integer n such that an=0and rnM(l(a))=anM;(iii) forany a, b∈R with0=ba∈N (R),there exists a positive integer n suchthat (ba)n=0andrM(Rb∩l((ab)n1a))=rM(b)+(ab)n1aM.(2)Let Rbe a ring,if every simple singular left R-module is Wnil-injective, thenthe following statements are equivalent:(i) R is reduced;(ii)R is left semicentral and N (R) forms a right ideal of R;(iii)R is a quasi ZI-ringand l(Ra)=Re, e~2=e∈R for any a∈R.