A Baer-Kaplansky theorem for modules over principal ideal domains

Breaz, Simion

doi:10.1216/jca-2015-7-1-1

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Cited by 5 publications

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“…[14,Example 9.2.3]. From a different perspective, it was proven in [2] that in the case of modules over principal ideal domains every module is determined up to isomorphism by the endomorphism ring of a convenient module. In particular, two abelian groups G and H are isomorphic if and only if the rings End(Z ⊕ G) and End(Z ⊕ H) are isomorphic.…”

Section: Introductionmentioning

confidence: 99%

The Baer-Kaplansky theorem for all abelian groups and modules

Breaz¹,

Brzeziński²

2021

Preprint

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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.

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Section: Introductionmentioning

confidence: 99%