Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

Torsion units in alternative loop rings


Authors: Edgar G. Goodaire and César Polcino Milies
Journal: Proc. Amer. Math. Soc. 107 (1989), 7-15
MSC: Primary 20N05; Secondary 17D05
MathSciNet review: 953005
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ {\mathbf{Z}}L$ denote the integral alternative loop ring of a finite loop $ L$. If $ L$ is an abelian group, a well-known result of $ {\text{G}}$. Higman says that $ \pm g,g \in L$ are the only torsion units (invertible elements of finite order) in $ {\mathbf{Z}}L$. When $ L$ is not abelian, another obvious source of units is the set $ \pm {\gamma ^{ - 1}}g\gamma $ of conjugates of elements of $ L$ by invertible elements in the rational loop algebra $ {\mathbf{Q}}L$. H. Zassenhaus has conjectured that all the torsion units in an integral group ring are of this form. In the alternative but not associative case, one can form potentially more torsion units by considering conjugates of conjugates $ \gamma _{^1}^{ - 1}\left( {\gamma _2^{ - 1}g{\gamma _2}} \right){\gamma _1}$ and so forth. In this paper we prove that every torsion unit in an alternative loop ring over $ {\mathbf{Z}}$ is $ \pm $ a conjugate of a conjugate of a loop element.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 20N05, 17D05

Retrieve articles in all journals with MSC: 20N05, 17D05


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1989-0953005-1
PII: S 0002-9939(1989)0953005-1
Keywords: Alternative ring, group ring, unit
Article copyright: © Copyright 1989 American Mathematical Society