Endomorphism rings via minimal morphisms

Abstract

We prove that if $u:K\to M$ is a left minimal extension, then there exists an isomorphism between two subrings, ${\text{End}}_R^M(K)$ and ${\text{End}}_R^K(M)$ of ${\text{End}}_R(K)$ and ${\text{End}}_R(M)$ 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 $M$.