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Normal Subloops in the Integral Loop Ring of an RA Loop

Published online by Cambridge University Press:  20 November 2018

Edgar G. Goodaire  and
César Polcino Milies
Edgar G. Goodaire
Affiliation:
Memorial University of Newfoundland St. John’s, Newfoundland A1C 5S7, email: [email protected]
César Polcino Milies
Affiliation:
Instituto de Matemática e Estatística Universidade de São Paulo Caixa Postal 66.281 CEP 05315-970 São Paulo SP Brasil, email: [email protected]
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Abstract

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We show that an $\text{RA}$ loop has a torsion-free normal complement in the loop of normalized units of its integral loop ring. We also investigate whether an $\text{RA}$ loop can be normal in its unit loop. Over fields, this can never happen.

Keywords

20N0517D0516S3416U60
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 44 , Issue 1 , 01 March 2001 , pp. 27 - 35
Copyright
Copyright © Canadian Mathematical Society 2001

References

[CG86] Chein, Orin and Goodaire, Edgar G., Loops whose loop rings are alternative. Comm. Algebra (2) 14 (1986), 293–310.Google Scholar
[dBJ97] de Barros, Luiz G. X. and Juriaans, Stanley O., Units in alternative integral loop rings. Resultate Math. 31 (1997), 266–281.Google Scholar
[GJM96] Goodaire, E. G., Jespers, E., and Milies, C. Polcino, Alternative loop rings, North-Holland Math. Studies 184, Elsevier, Amsterdam, 1996.Google Scholar
[GM89] Goodaire, Edgar G. and Milies, César Polcino, Torsion units in alternative loop rings. Proc. Amer.Math. Soc. 107 (1989), 7–15.Google Scholar
[GM95] Goodaire, Edgar G. and Milies, César Polcino, On the loop of units of an alternative loop ring. Nova J. AlgebraGeom. (3) 3 (1995), 199–208.Google Scholar
[GM96a] Goodaire, Edgar G. and Milies, César Polcino, Finite conjugacy in alternative loop algebras. Comm. Algebra (3) 24 (1996), 881–889.Google Scholar
[GM96b] Goodaire, Edgar G. and Milies, César Polcino, Finite subloops of units in an alternative loop ring. Proc. Amer.Math. Soc. (4) 124 (1996), 995–1002.Google Scholar
[Goo95] Goodaire, Edgar G., The radical of a modular alternative loop algebra. Proc. Amer. Math. Soc. (11) 123 (1995), 3289–3299.Google Scholar
[GP86] Goodaire, Edgar G. and Parmenter, M. M., Units in alternative loop rings. Israel J. Math. (2) 53 (1986), 209–216.Google Scholar
[GP87] Goodaire, Edgar G. and Parmenter, M. M., Semi-simplicity of alternative loop rings, Acta Math. Hungar. (3–4) 50 (1987), 241–247.Google Scholar
[Hig40] Graham Higman, The units of group rings. Proc. LondonMath. Soc. (2) 46 (1940), 231–248.Google Scholar
[JL93] Eric Jespers and Guilherme Leal, A characterization of the unit loop of the integral loop ring Z[M16 (Q, 2)]. J. Algebra (1) 155 (1993), 95–109.Google Scholar
[MZ] Milies, C. Polcino and Zatelli, Albertina, Nilpotent elements and ideals in alternative loop rings. East West J. Math., to appear.Google Scholar
[Pai56] Paige, Lowell J., A class of simple Moufang loops. Proc. Amer. Math. Soc. 7 (1956), 471–482.Google Scholar
[RS83] Roggenkamp, K. W. and Scott, L. L., Units in metabelian group rings: non-splitting examples for normalized units. J. Pure Appl. Algebra 27 (1983), 299–314.Google Scholar
[Seh78] Sehgal, S. K., Topics in group rings, Marcel Dekker, New York, 1978.Google Scholar
[Whi68] Whitcomb, A., The group ring problem, Ph.D. thesis, Chicago, 1968.Google Scholar
[ZSSS82] Zhevlakov, K. A., Slin’ko, A. M., Shestakov, I. P., and Shirshov, A. I., Rings that are nearly associative. Academic Press, New York, 1982, translated by Harry F. Smith.Google Scholar