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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. [ABSTRACT FROM AUTHOR]

A right R-module MR is called small injective if every homomorphism from a small right ideal to MR can be extended to an R-homomorphism from RR to MR. A ring R is called right small injective, if the right R-module RR is small injective. We prove that R is semiprimitive if and only if every simple right (or left) R-module is small injective. Further we show that the Jacobson radical J of a ring R is a noetherian right R-module if and only if, for every small injective module ER, E(ℕ) is small injective. [ABSTRACT FROM AUTHOR]