We define and investigate t-semisimple modules as a generalization of semisimple modules. A module M is called t-semisimple if every submodule N ? M contains a direct summand K of M such that K is t-essential in N. We will show in Theorem 2.3.3, Corollaries 2.3.7-2.3.11 and roposition 2.3.13, that t-semisimple modules admit many other characterizations. Submodules and homomorphic images of t-semisimple modules inherit the property, and every direct sum of t-semisimple modules is again a t-semisimple module. We will show that there exists a largest t-semisimple submodule in any module M , and (where is the Goldie torsion submodule and is the sum of nonsingular simple submodule ). It will be shown that a semilocal module M is t-semisimple if and only if Rad (M) is -torsion. The t-semisimple property is invariant under Morita equivalences. T-semisimple modules form a strict sub In Section 3 we defined and study right t-semisimple rings. A ring R is called right t-semisimple if is t-semisimple. Every right Artinian local ring is right t- semisimple. Various characterizations of right t-semisimple rings are given. Accordingly, a t-semisimple ring is exactly a direct product of two rings, one is semisimple and the other is right -torsion. For rings, being t-semisimple is not a symmetric property, and we give an example of a right t-semisimple ring which is not left t-semisimple. We determine rings R for which the R-modules coincide. For some types of rings , conditions equivalent to being t-semisimple are found. ection 3.2 concerns with other characterizations of right t-semisimple rings relative to chain conditions. It is shown that a ring R is right t-semisimple , if and only if, every nonsingular R-module has ACC (ascending chain condition) , respectively, DCC (descending chain condition) on essential submodules. Moreover, we see that every quasi-frobenius ring is right t-semisimple. For rings, we have quasi-frobenius t-semisimple = right ?-t -extending but none of these implications is reversible. . Finally, we give an example of a ring R for which every nonsingular cyclic R-module is injective but it is not right t-semisimple, that is, not every nonsingular R-module is injective.