On Σ-q rings

Nakayama [T. Nakayama, On Frobeniusean algebras II, Annals of Mathematics 42 (1941) 1-21] showed that over an artinian serial ring every module is a direct sum of uniserial modules. Hence artinian serial rings have the property that each right (left) ideal is a finite direct sum of quasi-injective right (left) ideals. A ring with the property that each right (left) ideal is a finite direct sum of quasi-injective right (left) ideals will be called a right (left) Σ-q ring. For example, commutative self-injective rings are Σ-q rings. In this paper, various classes of such rings that include local, simple, prime, right non-singular right artinian, and right serial, are studied. Prime right self-injective right Σ-q rings are shown to be simple artinian. Right artinian right non-singular right Σ-q rings are upper triangular block matrix rings over rings which are either zero rings or division rings. In general, a Σ-q ring is not left-right symmetric nor is it Morita invariant.