Let R be a commutative ring with 1 ≠ 0 and M a unital R-module. M is said to satisfy Property (A) if for each finitely generated ideal J of R contained in ZR(M) , there exists 0≠m∈M such that Jm=(0). Also M is said to satisfy Property (T) if for each finitely generated submodule N of M contained in T(M), there exists 0≠a∈R such that aN=(0). In this article, we study certain annihilator conditions on modules such as Property (A) and Property (T). In addition to give many properties of modules satisfying Property (A) (Property (T)), we characterize these classes of modules in terms of r-submodules and sr-submodules. Also, we give a method to construct non Noetherian rings in which every ideal satisfies Property (A).