Endomorphism rings via minimal morphisms

TL;DR

This paper establishes an isomorphism between certain subrings of endomorphism rings induced by minimal morphisms, enabling the transfer of properties between endomorphism rings of related modules.

Contribution

It introduces a new isomorphism between subrings of endomorphism rings associated with minimal extensions, linking their properties in novel ways.

Findings

Isomorphism between $ extrm{End}_R^M(K)$ and $ extrm{End}_R^K(M)$ modulo Jacobson radicals.

Application of the isomorphism to infer properties of endomorphism rings.

Conditions under which invariance under endomorphisms or automorphisms holds.

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 $M$.