Notes on quasi-Frobenius rings

Abstract

Abstract 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 ) {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 {R/S_{l}} is left Goldie, (5) R is a right YJ-injective right minannihilator ring with ACC on right annihilators.