On The Generalizations Of Right P-Injective

In the first section of this paper, we generalize right p-injective into right pm-injective. Firstly, the relationship between right pm-injective and various generalizations of right p-injective were discussed. It obtain that right pm-injective is not right p-injective, right pm-injecitve is not right GP-injective, and right pm-injective is right mininjecitve, but the conversely is not true. Secondly, the equivalence proposition of right pm-injective ring is obtained, and if R is right pm-injective ring, then the Jacobson radical of R coincide with the right singular ideal. Finally, it shows that R is von Neumann regular ring if and only if every right .R-module is right pm-injective if and only if every right principal right ideal of R is right pm-injective if and only if every essential right ideal of R is right pm-injective. Furthermore, some good properties are obtained.In the second section, we extend the conception of right pm-injecitve from ring to module and obtain the notion of right qpm-injective module. Firstly, an equivalence proposition of right qpm-injective module is discussed, and when the endomorphism of right qpm-injective module is right pm-injective ring is observed. The result is that: (1) A right R-module M is right qpm-injective module if and only if for any s.t S = End(M) with ker(s) = ker(t), then Ss = St. (2) if M is self-generator, then the endomorphism of right qpm-injective module M is right pm-injective ring. Finally, the properties of the endomorphism of right qpm-injective module are discussed and obtain some applications.In the third section, we generalize right pm-injective to right gpm-injective.It is easy to know that right gpm-injective is a generalization of right GP-injective, too. Firstly, the relationship between right gpm-injective and various generalizations of right p-injective were discussed. It obtain that right gpm-injecitve is not right GP-injective, and right pm-injective is right mininjecitve, but the conversely is not true. Secondly, the basic properties of right gpm-injective ring are discussed. Finally, it investigate QF-ring with right gpm-injective, and obtain the following theorem:Theorem The following statements are equivalent:(1) R is quasi-Frobeniusean ring:(2) R is right noetherian, left pm-injective ring, right pm-injective ring;(3) R is right noetherian, left pm-injective ring, right GP-injective ring;(4) R is right noetherian, left gpm-injective ring, right gpm-injective ring;(5) R is right noetherian, right gpm-injective ring and each right minimal ideal is right annihilator.