On the decomposition of nonsingular CS-modules

It is shown that if M is a nonsingular CS-module with an indecomposable decomposition M = +(i is an element of I) M(i), then the family {M(i)/i is an element of I} is locally semi-T-nilpotent. This fact is used to prove that any nonsingular self-generator C-CS module is a direct sum of uniserial Noetherian quasi-injective submodules. As an application, we provide a new proof of Goodearl's characterization of non-singular rings over which all nonsingular right modules are projective.