New characterizations of quasi-Frobenius rings

In this paper, we firstly provide several new characterizations of quasi-Frobenius rings by using some generalized injectivity of rings with certain chain conditions. For example, (1) a ring R is quasi-Frobenius if and only if R is right C-11, right minfull with ACC on right annihilators; (2) a ring R is quasi-Frobenius if and only if R is two-sided min-CS with ACC on right annihilators in which Soc(R-R) <= e R-R; (3) a ring R is quasi-Frobenius if and only if R is right Johns left C-11; (4) a ring R is quasi-Frobenius if and only if R is quasi-dual two-sided C-11 with ACC on right annihilators. Moreover, it is shown that a ring R is quasi-Frobenius if and only if R is a left P-injective left IN-ring with right RMC and Z(R-R) = Z(R-R). Also, we prove that if R is a right duo, right QF-3(+) left quasi-duo ring satisfying ACC on right annihilators, then R is quasi-Frobenius. In this paper, several known results on quasi-Frobenius rings are reproved as corollaries.