E-small essential submodules

Let R be a commutative ring with identity, and UR be an R-module, with E=End(UR). In this work we consider a generalization of class small essential submodules namely E-small essential submodules. Where the submodule Q of UR is said E-small essential if Q ∩W=0 , when W is a small submodule of UR, implies that NS(W)=0, where NS(W)={ψ∈E | Imψ⊆W}. The intersection B¯¯¯¯R(U) of each submodule of UR contained in Soc(UR). The B¯¯¯¯R(U) is unique largest E-small essential submodule of UR, if UR is cyclic. Also in this paper we study B¯¯¯¯R(U) and W¯¯¯¯¯E(U). The condition when B¯¯¯¯R(U) is E-small essential, and Tot( U,U)=W¯¯¯¯¯E(U)=J(E) are given.