Some properties of Z-small prime modules

Let R be a commutative ring with identity, and H be a unital (left) E-module. In this paper, we give a new properties of Z-small modules. Where an E-module H is a Z-small prime module if and only if ann H = ann K, for every non-zero submodule K of H such that K ≪_Z H. Where a submodule K of an E-module H is called Z-small (briefly K ≪ _Z H) if K+B=H with B ⊇ Z_2 (E) and B is a submodule of H, then B =H. Among of these properties if H is finitely generated faithful multiplication E-module, then H is a small Z-small prime E-module if and only if E is a Z-small prime ring. Also, we prove that an E-module H is a Z-small prime if and only if E-module the E-module E / (ann H) is cogenerated by every non-trivial Z-small submodule of H.