Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Trace functions in the ring of fractions of polycyclic group rings


Author: A. I. Lichtman
Journal: Trans. Amer. Math. Soc. 330 (1992), 769-781
MSC: Primary 16S34; Secondary 20C07
DOI: https://doi.org/10.1090/S0002-9947-1992-1040264-7
MathSciNet review: 1040264
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ KG$ be the group ring of a polycyclic-by-finite group $ G$ over a field $ K$ of characteristic zero, $ R$ be the Goldie ring of fractions of $ KG$, $ S$ be an arbitrary subring of $ {R_{n \times n}}$. We prove that the intersection of the commutator subring $ [S,S]$ with the center $ Z(S)$ is nilpotent. This implies the existence of a nontrivial trace function in $ {R_{n \times n}}$.


References [Enhancements On Off] (What's this?)

  • [1] R. L. Snider, The division ring of fractions of a group ring, Lecture Notes in Math., vol. 1029, Springer-Verlag, 1983, pp. 325-339. MR 732482 (85g:16009)
  • [2] A. Hattori, Rank element of a projective module, Nagoya J. Math. 15 (1965), 113-120. MR 0175950 (31:226)
  • [3] J. Stallings, Centerless group--an algebraic formulation of Gottlieb's theorem, Topology 4 (1965), 129-134. MR 0202807 (34:2666)
  • [4] D. S. Passman, The algebraic structure of group rings, Wiley-Interscience, New York, 1977. MR 470211 (81d:16001)
  • [5] A. I. Lichtman, On normal subgroups of the multiplicative group of skew fields generated by a polycyclic-by-finite group, J. Algebra 78 (1982), 548-577. MR 680374 (84f:16016)
  • [6] -, On linear groups of a field of fractions of a polycyclic group ring, Israel J. Math. 42 (1982), 318-326. MR 682316 (84d:20050)
  • [7] M. Shirvani and B. A. F. Wehrfritz, Skew linear groups, Cambridge Univ. Press, Cambridge, 1986. MR 883801 (89h:20001)
  • [8] B. A. F. Wehrfritz, On division rings generated by polycyclic groups, Israel J. Math. 47 (1984), 154-164. MR 738166 (85e:20037)
  • [9] C. Curtis and I. Reiner, Methods of representation theory, vol. I, Wiley-Interscience, New York, 1981. MR 632548 (82i:20001)
  • [10] P. Schmid, Lifting modular representations of $ p$-solvable groups, J. Algebra 83 (1983), 461-470. MR 714256 (85d:20009)
  • [11] P. M. Cohn, Free rings and their relations, Academic Press, New York, 1985. MR 800091 (87e:16006)
  • [12] D. S. Passman, Universal fields of fractions for polycyclic group algebras, Glasgow Math. J. 23 (1982), 103-113. MR 663135 (84a:16021)
  • [13] N. Jacobson, Lie algebras, Wiley-Interscience, New York, 1962. MR 0143793 (26:1345)
  • [14] A. Eisenbud and A. I. Lichtman, On embedding of group rings of residually torsion free nilpotent groups into skew fields, Trans. Amer. Math. Soc. 299 (1987), 373-385. MR 869417 (88a:16022)
  • [15] A. I. Lichtman, On $ PI$-subrings of matrix rings over some classes of skew fields, J. Pure Appl. Algebra 52 (1988), 77-89. MR 949338 (89i:16008)
  • [16] M. Lorenz, Division algebras generated by finitely generated nilpotent groups, Proc. NATO ASI, Methods in Ring Theory, Reidel, Boston, Mass., 1984, pp. 265-280. MR 725089 (85i:16014)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 16S34, 20C07

Retrieve articles in all journals with MSC: 16S34, 20C07


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1992-1040264-7
Article copyright: © Copyright 1992 American Mathematical Society

American Mathematical Society