Rings whose cyclics are C3-modules

It is well known that if every cyclic right module over a ring is injective, then the ring is semisimple artinian. This classical theorem of Osofsky promoted a considerable interest in the rings whose cyclics satisfy a certain generalized injectivity condition, such as being quasi-injective, continuous, quasi-continuous, or CS. Here we carry out a study of the rings whose cyclic modules are C3-modules. The motivation is the observation that a ring R is semisimple artinian if and only if every 3-generated right R-module is a C3-module. Many basic properties are obtained for the rings whose cyclics are C3-modules, and some structure theorems are proved. For instance, it is proved that a semiperfect ring has all cyclics C3-modules if and only if it is a direct product of a semisimple artinian ring and finitely many local rings, and that a right self-injective regular ring has all cyclics C3-modules if and only if it is a direct product of a semisimple artinian ring, a strongly regular ring and a 2 x 2 matrix ring over a strongly regular ring. Applications to the rings whose 2-generated modules are C3-modules, and the rings whose cyclics are ADS or quasi-continuous are addressed.