Words periodic over the center of a division ring
HTML articles powered by AMS MathViewer
- by Leonid Makar-Limanov and Peter Malcolmson PDF
- Proc. Amer. Math. Soc. 93 (1985), 590-592 Request permission
Abstract:
In generalization of a result of Herstein, the authors prove that, in a division ring with uncountable center, if any given nontrivial group word takes only values periodic over the center, then the division ring is commutative. Techniques include use of the result that a noncommutative division ring finite-dimensional over its center includes a nonabelian free group in its multiplicative group.References
-
J. L. Gonsalves, Free groups in subnormal subgroups and the residual nilpotence of the group of units of rings, Trab. Departamento Mat., Vol. 58, Univ. Sao Paulo, 1983.
- I. N. Herstein, Multiplicative commutators in division rings. II, Rend. Circ. Mat. Palermo (2) 29 (1980), no. 3, 485–489 (1981) (English, with Italian summary). MR 638685, DOI 10.1007/BF02849763
- Nathan Jacobson, Structure of rings, Revised edition, American Mathematical Society Colloquium Publications, Vol. 37, American Mathematical Society, Providence, R.I., 1964. MR 0222106
- Irving Kaplansky, A theorem on division rings, Canad. J. Math. 3 (1951), 290–292. MR 42389, DOI 10.4153/cjm-1951-033-7
- A. I. Lichtman, Free subgroups of normal subgroups of the multiplicative group of skew fields, Proc. Amer. Math. Soc. 71 (1978), no. 2, 174–178. MR 480623, DOI 10.1090/S0002-9939-1978-0480623-2
- A. Lichtman, On subgroups of the multiplicative group of skew fields, Proc. Amer. Math. Soc. 63 (1977), no. 1, 15–16. MR 447432, DOI 10.1090/S0002-9939-1977-0447432-0
- L. Makar-Limanov, On free subsemigroups of skew fields, Proc. Amer. Math. Soc. 91 (1984), no. 2, 189–191. MR 740167, DOI 10.1090/S0002-9939-1984-0740167-X
- W. R. Scott, On the multiplicative group of a division ring, Proc. Amer. Math. Soc. 8 (1957), 303–305. MR 83984, DOI 10.1090/S0002-9939-1957-0083984-8
- J. Tits, Free subgroups in linear groups, J. Algebra 20 (1972), 250–270. MR 286898, DOI 10.1016/0021-8693(72)90058-0
Additional Information
- © Copyright 1985 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 93 (1985), 590-592
- MSC: Primary 16A39; Secondary 16A70
- DOI: https://doi.org/10.1090/S0002-9939-1985-0776184-4
- MathSciNet review: 776184