Strongly t-continuous and Strongly t-semisimple Modules

Abstract

We introduce and investigate strongly t-continuous modules. A module   is called strongly t-continuous if  is strongly t-extending, and every submodule of  which contains  and is isomorphic to a direct summand is a fully invariant direct summand of . It is shown that, while a direct summand of strongly t-continuous inherits the property, a direct sum of strongly t-continuous modules don’t.  is strongly t-continuous  if and only if  is strongly t-extending and the endomorphism  ring of   is Von Neumann regular, if and only if , where  is a strongly t-continuous module. We have shown that strongly t-continuous module and t-continuous module are coinciding under certain conditions. Many other properties and example are given.