On coclosed submodules

Let R be a ring and let M be a right R-module. M is called a T-module if M/A ≅ M/B, where A is a coclosed submodule of M and B is any submodule of M, implies that B is a coclosed submodule of M. In this note we introduce T-modules to characterize when any finite direct sum of lifting/discrete modules is discrete. Moreover, we prove that any amply supplemented module is discrete if and only if it is a ⊕-supplemented T-module. Let M = M₁ ⊕... ⊕ M_n be an amply supplemented module. We prove that M is a T-module if and only if every M_i is a T-module and M ₁, ..., M_n are relatively projective.