A note on generalizations of quasi-Frobenius rings

Abstract

A ring [Formula: see text] is called quasi-Frobenius, briefly QF, if [Formula: see text] is right (or left) Artinian and right (or left) self-injective. A ring [Formula: see text] is called right co-Harada if every noncosmall right [Formula: see text]-module contains a nonzero projective direct summand and [Formula: see text] satisfies the ACC on right annihilators. The class of co-Harada rings is one of the most interesting generalizations of QF rings. When considering relation between these ring classes, [K. Oshiro, Lifting modules, extending modules and their applications to QF-ring, Hokkaido Math. J. 13 (1984) 310–338, Theorem 4.3] showed that a ring [Formula: see text] is QF if and only if it is right co-Harada ring with [Formula: see text]. In this note, we show that a ring [Formula: see text] is QF if and only if it is right co-Harada ring and satisfies either [Formula: see text] or [Formula: see text].