The Baer-Kaplansky theorem for almost completely decomposable groups

Dualities between Almost Completely Decomposable Groups and their Endomorphism Rings

“…Direct decomposition properties of acd groups are reflected in (or determined by) their endomorphism rings, which were studied by Mader and Schultz [16]. A tight connection between acd groups and their endomorphism rings was established in the case of block-rigid crq groups [6]. For a wider class of acd groups, we prove the following fact: if two acd groups of ring type are nearly isomorphic, then their endomorphism rings are also nearly isomorphic as Abelian groups.…”

“…The Baer-Kaplansky theorem was proved in [6] for block-rigid crq-groups up to near-isomorphism. We showed that two such groups X and Y of ring type are nearly isomorphic if and only if End X ∼ = End Y .…”

A Baer-Kaplansky theorem for modules over principal ideal domains

“…It is well known that Baer-Kaplansky Theorem cannot be extended to torsion-free groups (of rank 1) since there are infinitely many pairwise non-isomorphic torsion-free groups of rank 1 whose endomorphism rings are isomorphic to Z, [1]. However, similar results to Baer-Kaplansky Theorem hold for some special classes of torsion-free groups, see [2]. In the setting of modules over complete valuation domains W. May proved a theorem, [15,Theorem 1], for reduced modules which are neither torsion nor torsion-free and have a nice subgroup B such that M/B is totally projective: if M is such a module and N is an arbitrary module such that they have isomorphic endomorphism rings then M ∼ = N .…”

Almost completely decomposable groups with primary regulator quotients and their endomorphism rings

“…In [18, 15.2] and [8], it was shown that if X is an acd group, then End X is also an acd group as an additive structure.…”

“…Since endomorphism rings of acd groups are also acd groups (see [8,Proposition 3.1], [19,Lemma 3.1]), it is of great interest to clear up their group structure and obtain their typical acd group characteristics. Following this, the regulator of the additive group End X + of the endomorphism ring was completely described in [7, 3.3,3,4] for an arbitrary block-rigid acd group X with primary regulator quotient.…”