Modules for which every t-coclosed submodule has a supplement which is a summand

Let R be a ring and M an R-module. In this paper the notion of a t-small submoduleis used frequently. A submodule N of M is t-small in M (denoted by N t M), if N + K Z2(M) implies K Z2(M). Also N is called t-coclosed in M, in case N=K t M=K, for a submodule K contained in N, implies that N = K. We say that M is t 􀀀 D11 if every t-coclosed submodule of M has a supplement which is a direct summand of M. For an amply supplemented module M, we prove that M is t 􀀀 D11 if and only if M = Z2(M) L where Z2(M) and L are both t 􀀀 D11. Some relations of t 􀀀 D11-modules with the other types of modules investigated. Examples are presented to separate di erent concepts.