Some results of QF-\(n\) rings \((n=2,3)\) (Q1184870)

The main results of the paper are the following: Theorem 1: For a right Artinian ring \(R\) the following statements are equivalent: (i) \(R\) is QF (= Quasi-Frobenius ring), (ii) \(\operatorname{Soc}R_R\subset\operatorname{Soc}{_R R}\) and \(R\) is QF-2, (iii) \(\operatorname{Soc}R_R\subset\operatorname{Soc}{_R R}\) and \(R_R\oplus R_R\) or \({_R R}\oplus{_R R}\) has the property that for every submodule \(A\) of \(R_R\oplus R_R\) (resp. \({_R R}\oplus{_R R}\)) there exists a submodule \(A^*\) such that \(A\) is an essential submodule of \(A^*\) and \(A^*\) is a direct summand of \(R_R\oplus R_R\) (resp. \({_R R}\oplus{_R R}\)), which improves a result of the reviewer [J. Reine Angew. Math. 201, 100--112 (1959; Zbl 0094.25101)]. Theorem 2: A right or left Artinian ring \(R\) is QF-3 if \(R\) is QF-2.