Abstract
It is shown that if a division ring is such that for every element a there exist co-monic integer polynomials f a , G a of different orders so that f a(a)/G a(a) is central then the ring is commutative. We extend this result to reduced PI rings, and show that it breaks down slightly for primitive rings, where the exceptions are characterized. This extends some earlier theorems of Herstein and Faith.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
S. A. Amitsur, Generalized polynomial identities and pivotal monomials, Trans. Amer. Math. Soc. 114 (1965), 210–226.
V. A. Andrunakievic and J. M. Rjabuhin, Rings without nilpotent elements and completely prime ideals, Soviet Math. Doklady 9 (1968). 565–568.
C. Faith, Algebraic division ring extensions, Proc. Amer. Math. Soc. 11 (1960), 43–45.
C. Faith, Radical extensions of rings, Proc. Amer. Math. Soc. 12 (1961), 274–283.
E. S. Golod and I.R. Shafarevitch, On towers of class fields, Izv. Akad. Nauk. SSSR, Ser Math. 28 (1964), 261–272.
J. Z. Gonçalves Free groups in subnormal subgroups and the residual nilpotence of the group of units of group rings Canad. Math. Bull. 27 (1984), 365–370.
J. Z. Gonçalves, L. M. V. Figueiredo and M. Shirvani Free group algebras in certain division rings, (preprint).
J. Z. Gonçalves and A. Mandel, Are there free groups in division rings?, Israel J. of Math. 53 (1986), 69–80.
I. N. Herstein, The structure of certain classes of rings, Amer. J. of Math. 75 (1953), 864–871.
I. N. Herstein, Two remarks on commutativity of rings, Canad. J. of Math. 7 (1955), 411–412.
I. N. Herstein, On a result of Faith, Canad. Math. Bull. 18 (1975), 609.
I. N. Herstein, Rings with involution, Univ. of Chicago Press, Chicago, 1976.
I. Kaplansky, A theorem on division rings, Canad. J. of Math. 3 (1951), 290–292.
A. I. Lichtman, Free subgroups of normal subgroups of the multiplicative group of skew fields, Proc. Amer. Math. Soc. 71 (1978), 174–178.
A. I. Lichtman, On normal subgroups of the multiplicative group of skew fields generated by a polycyclic-by-finite group, J. of Algebra 78 (1982), 548–577.
A. I. Lichtman, On matrix rings and linear groups over a field of fractions of enveloping algebras and group rings, I J. of Algebra 88 (1984), 1–37.
L. Makar-Limanov, On Free Subobjects of Skew Fields, in: Methods in Ring Theory, Proceedings NATO ASI, Antwerp 1983 (F. van Oystaeyen ed.), NATO ASI series (C) vol. 129, pp. 281–286.
L. Makar-Limanov, The skew field of fractions of the Weyl algebra contains a free noncommutative subalgebra, Comm. Algebra 17 (1983), 2003–2006.
L. Makar-Limanov, On group rings of nilpotent groups, Israel. J. of Math. 48 (1984), 245–248.
L. Makar-Limanov and P. Malcolmson, Free subalgebras of enveloping algebras, Proc. Amer. Math. Soc. 111 (1991), 315–322.
M. Nagata, T. Nakayama and T. Tuzuku, An existence lemma in valuation theory, Nagoya Math. J. 6(1953), 59–61.
A. R. Ichoux, A commutativity theorem for division rings, Canad. Math. Bull. 23 (1980), 241–243.
A. Rosenberg and D. Zelinsky, On Nakayama’s extension of the x n(x) theorems, Proc. Amer. Math. Soc. 5 (1954), 484–486.
Author information
Authors and Affiliations
Corresponding author
Additional information
Work of both authors partially supported by CNPq
The second author was also supported by FAPESP
Rights and permissions
About this article
Cite this article
Gonçalves, J.Z., Mandel, A. A commutativity theorem for division rings and an extension of a result of Faith. Results. Math. 30, 302–309 (1996). https://doi.org/10.1007/BF03322197
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF03322197