LAUSR

Let $M$ be a right $R$-module and $S=End_R(M)$. We call $M$ a $\mathcal{K}$-extending module if for every element $\phi\in S$, Ker$\phi$ is essential in a direct summand of $M$. In this paper we investigate these modules. We give a characterization of $\mathcal{K}$-extending modules. We prove that if $M$ is a projective self-generator module, then $M$ is a $\mathcal{K}$-extending module and every finitely generated projective right ideal of $S$ is a summand if and only if $S$ is semiregular and $\Delta(M)=Jac(S)$, where $\Delta(M)=\{f\in S \mid Ker f\leq^e M \}$ if and only if $M$ is $Z(M)$-$\mathcal{I}$-lifting.