New characterizations of quasi-Frobenius rings

Abstract

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, [Formula: see text] a ring [Formula: see text] is quasi-Frobenius if and only if [Formula: see text] is right [Formula: see text], right minfull with ACC on right annihilators; [Formula: see text] a ring [Formula: see text] is quasi-Frobenius if and only if [Formula: see text] is two-sided min-CS with ACC on right annihilators in which [Formula: see text]; [Formula: see text] a ring [Formula: see text] is quasi-Frobenius if and only if [Formula: see text] is right Johns left [Formula: see text]; [Formula: see text] a ring [Formula: see text] is quasi-Frobenius if and only if [Formula: see text] is quasi-dual two-sided [Formula: see text] with ACC on right annihilators. Moreover, it is shown that a ring [Formula: see text] is quasi-Frobenius if and only if [Formula: see text] is a left [Formula: see text]-injective left IN-ring with right RMC and [Formula: see text]. Also, we prove that if [Formula: see text] is a right duo, right QF-[Formula: see text] left quasi-duo ring satisfying ACC on right annihilators, then [Formula: see text] is quasi-Frobenius. In this paper, several known results on quasi-Frobenius rings are reproved as corollaries.