On General Injective Rings with Chain Conditions

Abstract

Kupisch proved that if R is a left and right artinian QF-2 ring and Sr = Sl, then R is QF. A weaker condition for a ring to be a QF ring was obtained by Dan and Thuyet. They proved that if R is a right artinian QF-2 ring and Sr ≤ Sl, then R is QF. In this paper, we prove that if R is a QF-2 ring satisfying ACC on right annihilators in which Sl ≤ eRR (e.g., Sr ≤ Sl with Sr ≤e RR), then R is QF. It is also proved that R is QF if and only if R is a left ef-extending, right continuous ring with ACC on right annihilators.