Hostname: page-component-848d4c4894-pftt2 Total loading time: 0 Render date: 2024-05-01T04:06:33.937Z Has data issue: false hasContentIssue false
  • English
  • Français

On Zero-divisors in Group Rings of Groups with Torsion

Published online by Cambridge University Press:  20 November 2018

S. V. Ivanov  and
Roman Mikhailov
S. V. Ivanov
Affiliation:
Department of Mathematics, University of Illinois, 1409 West Green Street, Urbana, IL 61801, USA e-mail: ivanov@math.uiuc.edu
Roman Mikhailov
Affiliation:
Steklov Mathematical Institute, Gubkina 8, Moscow, 119991, Russia e-mail: romanvm@mi.ras.ru Chebyshev Laboratory, St. Petersburg State University, 14th Line, 29b, Saint Petersburg, 199178 Russia
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Nontrivial pairs of zero-divisors in group rings are introduced and discussed. A problem on the existence of nontrivial pairs of zero-divisors in group rings of free Burnside groups of odd exponent $n\,\gg \,1$ is solved in the affirmative. Nontrivial pairs of zero-divisors are also found in group rings of free products of groups with torsion.

Keywords

20C0720E0620F0520F50Burnside groupsfree products of groupsgroup ringszero-divisors
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 57 , Issue 2 , 14 June 2014 , pp. 326 - 334
Copyright
Copyright © Canadian Mathematical Society 2014

Footnotes

The first author is supported in part by NSF grant DMS 09-01782. This research of the second author is supported by the Chebyshev Laboratory (Department of Mathematics and Mechanics, St. Petersburg State University) under RF Government grant 11.G34.31.0026

References

[1] Cohn, P. M., On the free product of associative rings. II. The case of (skew) fields. Math. Z. 73 (1960, 433–456.http://dx.doi.org/10.1007/BF01215516 Google Scholar
[2] Cohn, P. M., On the free product of associative rings. III. J. Algebra 8 (1968, 376–383.http://dx.doi.org/10.1016/0021-8693(68)90066-5 Google Scholar
[3] Dokuchaev, M. A. and Singer, M. L. S., Units in group rings of free products of prime cyclic groups. Canad. J. Math. 50 (1998, no. 2, 312–322.http://dx.doi.org/10.4153/CJM-1998-016-2 Google Scholar
[4] Gerasimov, V. N., The group of units of a free product of rings. (Russian) Mat. Sb. 134(176)(1987), no. 1, 42–65; translation in Math. USSR-Sb. 62 (1989, no. 1, 41–63.Google Scholar
[5] Ivanov, S. V., Strictly verbal products of groups and A. I. Mal’tsev's problem on operations over groups. (Russian) Trudy Moskov. Mat. Obshch. 54 (1992, 243–277, 279; translation in Trans. Moscow Math. Soc. 1993, 217–249.Google Scholar
[6] Ivanov, S. V., The free Burnside groups of sufficiently large exponents. Internat. J. Algebra Comput. 4 (1994, no. 1–2, 1–308.Google Scholar
[7] Ivanov, S. V., An asphericity conjecture and Kaplansky problem on zero divisors. J. Algebra 216 (1999, no. 1, 13–19.http://dx.doi.org/10.1006/jabr.1998.7756 Google Scholar
[8] Ivanov, S. V., On subgroups of free Burnside groups of large odd exponent. Illinois J. Math. 47 (2003, no. 1–2, 299–304.Google Scholar
[9] Ivanov, S. V., Embedding free Burnside groups in finitely presented groups. Geom. Dedicata 111 (2005, 87–105.http://dx.doi.org/10.1007/s10711-004-2826-8 Google Scholar
[10] Ivanov, S. V. The Kourovka Notebook: Unsolved problems in group theory. Eleventh ed., Russian Academy of Sciences Siberian Division, Institute of Mathematics, Novosibirsk, 1990.Google Scholar
[11] Kurosh, A. G., The theory of groups. Chelsea, New York, 1956.Google Scholar
[12] Ol’shanskii, A. Yu., On the Novikov-Adian theorem. (Russian) Mat. Sb. 118(160)(1982), no. 2, 203–235, 287.Google Scholar
[13] Ol’shanskii, A. Yu., The geometry of defining relations in groups. Nauka, Moscow, 1989; English translation Math. and tts Applications, Soviet series, 70, Kluwer Acad. Publ., 1991.Google Scholar