On tau-Extending Modules

Motivated by [2] and [6], we introduce a generalization of extending (CS) modules by using the concept of tau-large submodule which was defined in [9]. We give some properties of this class of modules and study their relationship with the familiar concepts of tau-closed, tau-complement submodules and the other generalization of extending modules (tau-complemented, tau-CS, s-tau-CS modules). We are also interested in determining when a tau-divisible module is tau-extending. For a tau-extending module M with C3, we obtain a decomposition theorem that there is a submodule K of M such that M = tau(M) circle plus K and K is tau (M)-injective. We also treat when a direct sum of tau-extending modules is tau-extending.