Modules whose closed submodules are finitely generated

Abstract

A module M is called a CC -module if every closed submodule of M is cyclic. It is shown that a cyclic module M is a direct sum of indecomposable submodules if all quotients of cyclic submodules of M are CC -modules. This theorem generalizes a recent result of B. L. Osofsky and P. F. Smith on cyclic completely CS -modules. Some further applications are given for cyclic modules which are decomposed into projectives and injectives.