On Some Classes of Artinian Rings

A module M is called a CS-module if every submodule of M is essential in a direct summand of M. A ring R is called CS-semisimple if every right R-module is CS. For a ring R, we show that:• is right artinian with Jacobson radical cube zero if every countably generated right -module is a direct sum of a projective module and a CS-module.• The following conditions are equivalent: (i) Every countably generated right -module is a direct sum of a projective module and a quasicontinuous module; and (ii) every right -module is a direct sum of a projective module and a quasi-injective module. We describe the structure of rings in (2) and show that such a ring is not necessarily CS-semisimple.