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.1980 Mathematics subject classification: Primary 16A52, 16A53.In [6,7] Osofsky proved that a ring R is semisimple Artinian if every cyclic right J?-module is injective. Since that time, this important theorem has been extensively investigated and extended by several authors (see e.g. The purpose of this note is to present some extensions of Osofsky-Smith's theorem in [8]. First we show that a cyclic module M is a direct sum of indecomposable submodules if all quotients of cyclic submodules of M have closed submodules cyclic. Our arguments use the idea of proof in [8]. A similar result holds also for finitely generated modules with closed submodules finitely generated. Further we prove that a finitely generated module M has finite uniform dimension if every quotient of a cyclic submodule of M is a direct sum of a projective module and a CS-module. As a consequence, we obtain a module-theoretic version of a result in [8] that right CDPIrings are right Noetherian. Finally, an application is given for right linearly topologized rings.

The Osofsky-Smith Theorem for Modular Lattices, and Applications (II)

“…Following Dung [19], a right R-module M is said to be CF if every closed submodule of M is finitely generated, and completely CF provided every quotient of M is also CF. Similarly, an object X of a Grothendieck category is called CF (acronym for Closed are Finitely generated) if every closed subobject of X is finitely generated, and completely CF if every quotient object of X is CF.…”

“…Corollary 4.4 is the categorical counterpart of the module-theoretical result Dung [19,Theorem 2.5], which in turn, can be viewed as a generalization of the Osofsky-Smith Theorem.…”

“…The author thanks N. V. Dung for drawing to his attention the paper Dung [19], Constantin Nȃstȃsescu and John Van Den Berg for helpful discussions, Septimiu Crivei for providing him references [13,14,16], and the referee for his/her careful reading of the manuscript and helpful suggestions. …”

A Research on the Generalizations of Modules Whose Submodules are Isomorphic to a Direct Summand

“…Since each 𝑀 * is 𝐶𝐻, for any 𝑖 ∈ 𝐼 there exists 𝑓 * ∈ 𝐸𝑛𝑑 % (𝑀 * ) such that 𝑓 * (𝑀 * ) = 𝐶 ∩ 𝑀 * . Now we can define the epimorphism Nguyen V. Dung[21] call a module 𝑀 𝐶𝐹 if any closed (complement) submodule is finitely generated. Now, it is proved that if a module 𝑀 is morphic and 𝐶𝐻 then 𝑀 is finitely generated if and only if 𝑀 is 𝐶𝐹.…”