Artinian Rings Characterized by Direct Sums of CS Modules#

Abstract

In this note, we show that the following are equivalent for a ring R for which the socle or the injective hull of R R is finitely generated: (i) The direct sum of any two CS right R-modules is again CS; (ii) R is right Artinian and every uniform right R-module has composition length at most two. Next we give partial answers to a question of Huynh whether a right countably Σ-CS ring which either is semilocal or has finite Goldie dimension is right Σ-CS. We give characterizations, in terms of radicals, of when such rings are right Σ-CS. In particular, for the semilocal case, Huynh's question is reduced to whether rad(Z 2(R R )) is Σ-CS or Noetherian, where Z 2(R R ) is the second singular right ideal of R. Our results yield new characterizations of QF-rings.