ANNIHILATOR-SMALL SUBMODULES

ABSTRACT. Let MR be a module with S = End(MR). We call a submodule K of Mr annihilator-small if K + T = M, T a submodule of MR, implies that is(T) = 0, where is indicates the left annihilator of T over S. The sum AR(M) of all such submodules of MR contains the Jacobson radical Rad(M) and the left singular submodule Zs(M). If MR is cyclic, then Ar(M) is the unique largest annihilator-small submodule of MR. We study ARM) and KS(M) in this paper. Conditions when AR(M) is annihilator-small and KS(M) = J(S) = Tot(M,M) are given.