Abstract

A ring R is called quasi-Frobenius, briefly QF, if R is right (or left) Artinian and right (or left) self-injective. A ring R is called right co-Harada if every noncosmall right R-module contains a nonzero projective direct summand and R 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 R is QF if and only if it is right co-Harada ring with Z(R)=J(R). In this note, we show that a ring R is QF if and only if it is right co-Harada ring and satisfies either Soc(RR)⊆Soc(RR) or Soc(RR)⊆Soc(RR).