On \(\Sigma\)-\(q\) rings. (Q1008744)

A ring is called a right \(\Sigma\)-\(q\) ring if every right ideal is a finite direct sum of quasi-injective right ideals. The authors study various classes of such rings, which include Artinian serial, commutative self-injective, local, simple, prime, right non-singular right Artinian, or right serial rings. They show that prime right self-injective right \(\Sigma\)-\(q\) rings are simple Artinian, and also, right Artinian right non-singular right \(\Sigma\)-\(q\) rings are upper triangular block matrix rings over rings which are either zero rings or division rings. In general, a \(\Sigma\)-\(q\) ring is neither left-right symmetric nor Morita invariant.