T-continuous modules

We introduce and investigate t-continuous modules. A module M is called t-continuous if M is t-extending, and every submodule of M which contains Z(2)(M) and is isomorphic to a direct summand of M, is itself a direct summand. The t-continuous property is inherited by direct summands. It is shown that M is a t-continuous module, if and only if, M is t-extending and the endomorphism ring of M/Z(2)(M) is von Neumann regular, if and only if, M = Z(2)(M) circle plus M', where M' is a continuous module. The rings R for which every (finitely generated, cyclic, free) R-module is t-continuous are characterized. It is proved that every t-continuous R-module is continuous exactly when R is a right SI-ring. Moreover, it is shown that the notions of a right GV-ring and a right V-ring coincide for right t-continuous rings.