Abstract
One of the continuity conditions identified by Utumi on self-injective rings is the C3-condition, where a module M is called a C3-module if whenever A and B are direct summands of M and A boolean AND B = 0, then A circle plus B is a summand of M. In addition to injective and direct-injective modules, the class of C3-modules includes the semisimple, continuous, indecomposable and regular modules. Indeed, every commutative ring is a C3-ring. In this paper we provide a general and unified treatment of the above mentioned classes of modules in terms of the C3-condition, and establish new characterizations of several well known classes of rings.