Notes on quasi-Frobenius rings

We give some new characterizations of quasi-Frobenius rings. Namely, we prove that for a ring R, the following statements are equivalent: (1) R is a quasi-Frobenius ring, (2) M-2(R) is right Johns and every closed left ideal of R is cyclic, (3) R is a left 2-simple injective left Kasch ring with ACC on left annihilators, (4) R is a left 2-injective semilocal ring such that R/S-l is left Goldie, (5) R is a right YJ-injective right minannihilator ring with ACC on right annihilators.