T-SEMISIMPLE MODULES AND T-SEMISIMPLE RINGS

We define and investigate t-semisimple modules as a generalization of semisimple modules. A module M is called t-semisimple if every submodule N contains a direct summand K of M such that K is t-essential in N. T-semisimple modules are Morita invariant and they form a strict subclass of t-extending modules. Many equivalent conditions for a module M to be t-semisimple are found. Accordingly, M is t-semisiple, if and only if, M=Z(2)(M) circle plus S(M) (where Z(2)(M) is the Goldie torsion submodule and S(M) is the sum of nonsingular simple submodules). A ring R is called right t-semisimple if R-R is t-semisimple. Various characterizations of right t-semisimple rings are given. For some types of rings, conditions equivalent to being t-semisimple are found, and this property is investigated in terms of chain conditions.