Endomorphism rings via minimal morphisms

Abstract: We prove that if $u:K \rightarrow M$ is a left minimal extension, then there exists an isomorphism between two subrings, $\textrm{End}_R^M(K)$ and $\textrm{End}_R^K(M)$ of $\textrm{End}_R(K)$ and $\textrm{End}_R(M)$ respectively, modulo their Jacobson radicals. This isomorphism is used to deduce properties of the endomorphism ring of $K$ from those of the endomorphism ring of $M$ in certain situations such us when $K$ is invariant under endomorphisms of $M,$ or when $K$ is invariant under automorphisms of $M$.