Bulletin of the
Korean Mathematical Society BKMS

This article mainly concentrates on the open question wheth\-er a right self-injective ring $R$ is necessary QF if $R/S_l$ is left Goldie. It is answered affirmatively under the condition $S_l\subseteq S_r$, where $S_l$ and $S_r$ denote the left socle and right socle of $R$ respectively. And the original condition ``right self-injective" can be weakened to ``right CS and right P-injective". It is also proved that a semiperfect, left and right mininjective ring $R$ is QF if $S_r\subseteq^{ess}R_R$ and $R/S_l$ is left Goldie.

Keywords: QF rings, annihilators, mininjective rings

MSC numbers: Primary 16L60, 16D50

Supported by: The research was supported by the Fundamental Research Funds for the Central Universities (No. 2242017K40085) and NSFC (No.11871145).