S-Essential Submodules

Let MR be a module with S = End(MR). We say that a submodule Kof a module MR is s-essential if K ∩ X = 0, X a submodule of MR, impliesthat DS(X) = 0 where DS(X) = {φ ∈ S | Imφ ⊆ X}. The intersectionA/bar/R(M) of all such submodules of MR is contained in Soc(M). If MR is finitelycogenerated, then A/bar/R(M) is the unique smallest s-essential submodule of MR.We study A/bar/R(M) and KS(M) in this paper. Conditions when A/bar/R(M) is sessential and K/bar/S(M) = J(S) = Tot(M, M) are given.