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Abelian groups that are torsion over their endomorphism rings

Published online by Cambridge University Press:  17 April 2009

J. Hill ,
P. Hill  and
W. Ullery
J. Hill
Affiliation:
Department of Mathematics, Yale University, New Haven, CT 06520, United States of America, e-mail: jhill@math.yale.edu
P. Hill
Affiliation:
Department of Mathematics, Auburn University, Auburn, AL 36849, United States of America, e-mail: hillpad@math.auburn.edu
W. Ullery
Affiliation:
Department of Mathematics, Auburn University, Auburn, AL 36849, United States of America, e-mail: ullery@math.auburn.edu
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Using Lambek torsion as the torsion theory, we investigate the question of when an Abelian group G is torsion as a module over its endomorphism ring E. Groups that are torsion modules in this sense are called ℒ-torsion. Among the classes of torsion and truly mixed Abelian groups, we are able to determine completely those groups that are ℒ-torsion. However, the case when G is torsion free is more complicated. Whereas no torsion-free group of finite rank is ℒ-torsion, we show that there are large classes of torsion-free groups of infinite rank that are ℒ-torsion. Nevertheless, meaningful definitive criteria for a torsion-free group to be ℒ-torsion have not been found.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 64 , Issue 2 , October 2001 , pp. 255 - 263
Copyright
Copyright © Australian Mathematical Society 2001

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