The Baer-Kaplansky theorem for all abelian groups and modules

TL;DR

This paper extends the Baer-Kaplansky theorem to all abelian groups and modules by replacing endomorphism rings with endomorphism trusses of heaps, showing these structures uniquely determine the groups and modules.

Contribution

It introduces the concept of endomorphism trusses of heaps to generalize the Baer-Kaplansky theorem to all abelian groups and modules.

Findings

Abelian groups are determined by their endomorphism trusses.

Isomorphisms of endomorphism trusses induce isomorphisms of groups.

Endomorphism trusses uniquely determine modules over rings.

Abstract

It is shown that the Baer-Kaplansky theorem can be extended to all abelian groups provided that the rings of endomorphisms of groups are replaced by trusses of endomorphisms of corresponding heaps. That is, every abelian group is determined up to isomorphism by its endomorphism truss and every isomorphism between two endomorphism trusses associated to some abelian groups $G$ and $H$ is induced by an isomorphism between $G$ and $H$ and an element from $H$. This correspondence is then extended to all modules over a ring by considering heaps of modules. It is proved that the truss of endomorphisms of a heap associated to a module $M$ determines $M$ as a module over its endomorphism ring.