On almost QF rings
A semiprimary ring \(R\) is called a right almost QF ring if \(R\) satisfies acc on right annihilators and every indecomposable projective right \(R\)-module is almost injective, and dually \(R\) is called a right almost co-QF ring if \(R\) satisfies dcc on right annihilators and every indecomposable injective right \(R\)-module is almost projective. Furthermore a semiprimary ring \(R\) is called a right \(H\) ring if \(R\) satisfies dcc on right annihilators and for a given non-small module \(M\), there exists a monomorphism from some nonzero injective module to \(M\). Dually a semiprimary ring \(R\) is called a right co-\(H\) ring if \(R\) satisfies acc on right annihilators and for a given non-cosmall module, there exists an epimorphism from \(M\) to some nonzero projective module. Without assuming that \(R\) is two-sided Artinian, it is shown that the following conditions are equivalent: (i) \(R\) is a right almost QF ring; (ii) \(R\) is a left almost co-QF ring; (iii) \(R\) is a right co-\(H\) ring; (iv) \(R\) is a left \(H\) ring. Also it is proved that under these conditions, the ring \(R\) is necessarily left Artinian.
scientific article; zbMATH DE number 1018457