On Three Kinds Of Continuous Modules

For a ring R,an R-module M is said to be a C1-module if every closed submodule of M is a direct summand of M.C1-modules are one of important kinds of modules which have been well studied.What's more,C1-modules have some generalizations such as C2-modules,C3-modules,C4-modules and so on.An R-module M is said to be a continuous module if M is both a C1-module and a C2-module.Similarly,an R-module M is said to be a quasi-continuous module if M is both a C1-module and a C3-module.An R-module M is said to be a pseudo-continuous module if M is both a C1-module and a C4-module.In this article,we mainly investigate these three kinds of modules.The structure of this article is as follows.The first three parts mainly introduce some basic definitions,lemmas and theorems.Moreover,many conclusions about some well-known modules like continuous modules and quasi-continuous modules will be given.At last,we will combine C4-modules and pseudo-continuous modules defined by N.Q.Ding et al.in 2017 with Goldie torsion submodules and give some generalizations.Some specific modules will be discussed in this part.