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