The Baer–Kaplansky Theorem for all abelian groups and modules

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 [Formula: see text] and [Formula: see text] is induced by an isomorphism between [Formula: see text] and [Formula: see text] and an element from [Formula: see text]. 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 [Formula: see text] determines [Formula: see text] as a module over its endomorphism ring.

Heaps of modules and affine spaces

“…) and hence Φ is well-defined by Corollary 4.3 again. The proof that it is an isomorphism of trusses is identical to the first part of the proof of [3,Theorem 3.3].…”

“…The Baer-Kaplansky Theorem for T -groups. This first subsection is entirely devoted to an application of the theory developed in this paper, which represents the natural extension of the results from [3] in view of what we proved in Section 4.…”

“…Thus we are led to consider also the full subcategory R-HMod of T(R)-HMod in which all objects are coming from R-modules, that is, T(R)-modules with exactly one absorber (see [9,Lemma 4.6 (2)(ii)]). The following is a particular instance of Corollary 4.3 (see also [3,Lemma 3.1]). Proposition 4.7.…”

“…Let R be a ring and let T(R) be the associated truss. A subset S of an R-module M is an equivalence class of a congruence modulo an R-submodule N if and only if it is an induced T(R)-submodule of the T(R)-module T(M) as in [9, §4] or Example 2.17 (3).…”

“…A recent attempt [3] to extend the Baer-Kaplansky theorem, relating isomorphisms of abelian p-groups to isomorphisms of their endomorphism rings [20,Theorem 16.2.5], to all abelian groups and to modules over rings led first to realise that the rings of group endomorphisms should be replaced by the trusses of the corresponding heap endomorphisms [6], and then that endomorphisms of modules should be replaced by endomorphisms of some more general module-like structures. The aim of the present text is to introduce and study in a systematic way such structures, which we term heaps of modules.…”