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Endomorphism Rings of Abelian Groups with Ample Divisible Subgroups

Endomorphism Rings of Abelian Groups with Ample Divisible Subgroups ENDOMORPHISM RINGS OF ABELIAN GROUPS WITH AMPLE DIVISIBLE SUBGROUPS WARREN MAY Let G and G* be abelian groups whose endomorphism rings E(G) and E(G*) are isomorphic. There are few theorems in the literature giving circumstances under which isomorphism of G with G* can be inferred. If we are dealing with torsion groups, then the well-known theorem of Baer and Kaplansky [3; Theorem 28] applies. Hauptfleisch [2] and Wolfson [7, 8] have obtained results for certain classes of torsion- free groups. The case of mixed abelian groups of torsion-free rank one has been considered by May and Toubassi [4, 5]. All of these results place conditions on both G and G*. There are fewer theorems that place restrictions only on G. As an example, the torsion groups G for which one can conclude that G is isomorphic to G* are determined in [6]. In this paper, we shall give a class of abelian groups such that if G is in the class, then isomorphism of E(G) with £(G*) implies isomorphism of G with G*. In studying the isomorphism problem, it is apparent that an abundance of available mappings increases the probability of obtaining positive results. Consequently, we wish to consider http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

Endomorphism Rings of Abelian Groups with Ample Divisible Subgroups

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Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/10.3.270
Publisher site
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Abstract

ENDOMORPHISM RINGS OF ABELIAN GROUPS WITH AMPLE DIVISIBLE SUBGROUPS WARREN MAY Let G and G* be abelian groups whose endomorphism rings E(G) and E(G*) are isomorphic. There are few theorems in the literature giving circumstances under which isomorphism of G with G* can be inferred. If we are dealing with torsion groups, then the well-known theorem of Baer and Kaplansky [3; Theorem 28] applies. Hauptfleisch [2] and Wolfson [7, 8] have obtained results for certain classes of torsion- free groups. The case of mixed abelian groups of torsion-free rank one has been considered by May and Toubassi [4, 5]. All of these results place conditions on both G and G*. There are fewer theorems that place restrictions only on G. As an example, the torsion groups G for which one can conclude that G is isomorphic to G* are determined in [6]. In this paper, we shall give a class of abelian groups such that if G is in the class, then isomorphism of E(G) with £(G*) implies isomorphism of G with G*. In studying the isomorphism problem, it is apparent that an abundance of available mappings increases the probability of obtaining positive results. Consequently, we wish to consider

Journal

Bulletin of the London Mathematical SocietyWiley

Published: Nov 1, 1978

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