Some properties of T-supplemented modules

Let M be an R-module and T≤ M. M is called a T-supplemented module if every submodule of M that contains T has a supplement in M. In this work, some properties of these modules are investigated. Let M be a T-supplemented R-module with T≤ M. If T is (cofinitely) supplemented, then M is (cofinitely) supplemented. If T M, then M is supplemented. Let M=M₁+M₂+...+Mₙ with Mᵢ≤ M ( i=1,2,...,n ) and T=T₁+T₂+...+Tₙ with Tᵢ≤ Mᵢ ( i=1,2,...,n ). If Mᵢ is Tᵢ-supplemented for each $ i=1,2,...,n$, then M is T-supplemented. Let M be an R-module and T≤ M. Then M is a T-supplemented if and only if M( ) is T( )-supplemented for every finite index set . Let f:M N be an R-module epimorphism and T≤ M. If M is T-supplemented, then N is f ( T )-supplemented. Let M be an R-module and T,V≤ M. If V is a supplement of a submodule U of M with T≤ U, then V is called a T-supplement submodule in M. Let M be an R-module and T≤ M. If every submodule of M that contain T is β equivalent to a T-supplement submodule in M, then M is T-supplemented.