Hostname: page-component-8448b6f56d-t5pn6 Total loading time: 0 Render date: 2024-04-24T21:03:52.863Z Has data issue: false hasContentIssue false
  • English
  • Français

A NOTE ON GROUP RINGS WITH TRIVIAL UNITS

Part of: Rings and algebras arising under various constructions Conditions on elements

Published online by Cambridge University Press:  19 July 2021

A. Y. M. CHIN Open the ORCID record for A. Y. M. CHIN [Opens in a new window]
A. Y. M. CHIN*
Affiliation:
Institute of Mathematical Sciences, Faculty of Science, Universiti Malaya, 50603Kuala Lumpur, Malaysia
*
e-mail: acym@um.edu.my
Get access Rights & Permissions [Opens in a new window]

Abstract

Let R be a ring with identity of characteristic two and G a nontrivial torsion group. We show that if the units in the group ring $RG$ are all trivial, then G must be cyclic of order two or three. We also consider the case where R is a commutative ring with identity of odd prime characteristic and G is a nontrivial locally finite group. We show that in this case, if the units in $RG$ are all trivial, then G must be cyclic of order two. These results improve on a result of Herman et al. [‘Trivial units for group rings with G-adapted coefficient rings’, Canad. Math. Bull.48(1) (2005), 80–89].

Keywords

group ringunittorsion grouplocally finite group

MSC classification

Primary: 16S34: Group rings , Laurent polynomial rings
Secondary: 16U60: Units, groups of units
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 105 , Issue 2 , April 2022 , pp. 243 - 247
Check for updates
Copyright
© 2021 Australian Mathematical Publishing Association Inc.

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Connell, I. G., ‘On the group ring’, Canad. J. Math. 15 (1963), 650–685.10.4153/CJM-1963-067-0CrossRefGoogle Scholar
Gardam, G., ‘A counterexample to the unit conjecture for group rings’, Preprint, 2021, arXiv:2102.11818v3.10.4007/annals.2021.194.3.9CrossRefGoogle Scholar
Herman, A. and Li, Y., ‘Trivial units for group rings over rings of algebraic integers’, Proc. Amer. Math. Soc. 134(3) (2006), 631–635.CrossRefGoogle Scholar
Herman, A., Li, Y. and Parmenter, M. M., ‘Trivial units for group rings with $G$ -adapted coefficient rings’, Canad. Math. Bull. 48(1) (2005), 80–89.10.4153/CMB-2005-007-1CrossRefGoogle Scholar
Higman, G., Units in Group Rings (D. Phil. Thesis, University of Oxford, 1940).10.1112/plms/s2-46.1.231CrossRefGoogle Scholar
Kaplansky, I., ‘Problems in the theory of rings’, Report of a Conference on Linear Algebras, June, 1956, National Academy of Sciences Publications, 502 (National Research Council, Washington, DC, 1957), 1–3.Google Scholar
Passman, D. S., The Algebraic Structure of Group Rings (Wiley–Interscience, New York–London–Sydney, 1977).Google Scholar
San Soucie, R. L., ‘Right alternative division rings of characteristic two’, Proc. Amer. Math. Soc. 6 (1955), 291–296.10.1090/S0002-9939-1955-0068526-3CrossRefGoogle Scholar