Right Pm-Injectivity And Correlation Study

In the first section of this paper,we generalize right pm-injective and right principally pseudo-injective into right M-pm-injective.Firstly,the definition and some basic properties of M-pm-injective are given.It also shows the relationship among right M-pm-injective,right pm-injective and principally pseudo-injective.Besides,we discuss some important properties of principally pseudo-injective modules and characterize semisimple Artinian ring With the aid of right pm-injective modules.As a consequence of this it is shown that:R is a semisimple Artinian ring if and only if every maximal essential right ideal of R is the right annihilator and every singular simple right R-module is pm-injective.A prime ring R is simple Artinian if and only if nonzero Soc(R)is pm-injective as a right R-module and R satisfies the ascending chain condition for special right annihilators.Then it introduces the concept of pm-injective dimension and obtains some results on pm-injective modules. Moreover,semi-simple ring,von Neumann regular ring and hereditary ring are characterized by pm-injective dimension.In the second section,we generalize right pm-injective to right n-pm-injective and give an equivalent characterization of n-pm-injective modules:M is a right n-pm-injective module if and only if A_a"=B_a".a". Furthermore,we give an equivalent characterization of n-regular rings in terms of right n-pm-injective modules.In the third section,we discuss the relationship between right gpm-injective and von Neumann regular ring,right gpm-injective and strongly regular ring respectively.It characters von Neumann regular rings with the aid of right gpm-injective modules.On the one hand,when R satisfies element right zero divison power,ring R is regularity if and only if every cyclic right R-module is right gpm-injective module if and only if every essential right ideal of R is right gpm-injective module.On the other hand,when R satisfies ACC of special right annihilator,R is von Neumann regular ring if and only if R is a semiprimitive right gpm-injective ring if and only if R is a right nonsingular right gpm-injective ring.Finally,it characterizes strongly regular rings via right gpm-injectivity and obtain some equivalent conditions.These effecitively improve many important results with regard to von Neumann regular rings.If R is a reduced ring,then R is a strongly regular ring if and only if L is a right pm-injective module for any L∈ME(R_R)if and only if L is a right gpm-injective module for any L∈ME(R_R).If every maximal right ideal of R is a weakly ideal,then R is a strongly regular ring if and only if R is a fully idempotent right gpm-injective ring and aL is right gpm-injective for any L∈ME(R_R)and a∈R if and only if R is a semiprime right gpm-injective ring and aL is right gpm-injective for any L∈ME(R_R)and a∈R.R is a strongly regular ring if and only if every maximal essential right ideal of R is right gpm-injective as a right R-module and R satisfies(*)condition.