Ap-injective Ring With Gp-injective Ring

In the first chapter of this paper , some characters of AP-injective ring are studied firstly . Then we genenalize strong left P-injective into strong left AP-injective , and obtain an important character : if R is a strong left AP-injective ring , then R/J(R) is a Von Neumann regular ring . Secondly , it is studied that when left AP-injective rings will be left continuous . Several conditions are given in this chapter , the main results are : when left AP-injective ring R is a left duo ring , and also is a local Baer ring ; or when left AP-injective ring R is a left weak injective ring ; or when left AP-injective ring R is left MI ring , etc, R can be a left continuous ring .In the second chapter , the chief content is the studies of the relationship between AGP-injective rings and regular rings . Firstly some characters of AGP-injective rings are discussed . And we obtain that right AGP-injective rings are right weak C₂ ring ; and right quasi-continuous right AGP-injective rings are right continuous rings . Meanwhile we prove that if L is a maximal right annihilator of semi-prime right AGP-injective ring R , then : (1) L is generated by idempotents ; (2) L is a maximal right ideal of R . Secondly , the equivalent conditions of AGP-injective rings and regular rings are studied , and Ï€^*- regular ring and G*PP ring are defined . The main results we obtained are that : R is a Von Neumann regular ring if and only if R is a right G*PP right AGP-injective ring ; if R is a right non-singular right AGP-injective ring , and l(I∩ J) = l(I) + l(J) , in which I, J are any right ideals of R , then R is a Von Neumann regular ring ; if R is semi-prime ERT, right AGP-ring , and Soc(R_r) = 0 , then R is a strong regular ring , and so on . At last , we genenalize n-P-injective into n-AP-injective , and discuss its characters . Then we obtain that right n-PP right n-AP-injective ring R is n-regular ring ; if reduced right n-AP-injective ring R is a local ring , and Ra = aR for any a ∈ R, then U C J(R) or U is a summand of R for any uniform...