On embedding of group rings of residually torsion free nilpotent groups into skew fields

Eizenbud, A.; Lichtman, A. I.

doi:10.1090/S0002-9947-1987-0869417-3

On embedding of group rings of residually torsion free nilpotent groups into skew fields
HTML articles powered by AMS MathViewer

by A. Eizenbud and A. I. Lichtman

Trans. Amer. Math. Soc. 299 (1987), 373-386

DOI: https://doi.org/10.1090/S0002-9947-1987-0869417-3

PDF | Request permission

Abstract:

It is proven that the group ring of an amalgamated free product of residually torsion free nilpotent groups is a domain and can be embedded in a skew field. This is a generalization of J. Lewin’s theorem, proven for the case of free groups. Our proof is based on the study of the Malcev-Neumann power series ring $K\left \langle G \right \rangle$ of a residually torsion free nilpotent group $G$. It is shown that its subfield $D$, generated by the group ring $KG$, does not depend on the order of $G$ for many kinds of orders and the study of $D$ can be reduced in some sense to the case when $G$ is nilpotent.

References

Jacques Lewin, Fields of fractions for group algebras of free groups, Trans. Amer. Math. Soc. 192 (1974), 339–346. MR 338055, DOI 10.1090/S0002-9947-1974-0338055-4
P. M. Cohn, On the free product of associative rings. III, J. Algebra 8 (1968), 376–383. MR 222118, DOI 10.1016/0021-8693(68)90066-5
P. M. Cohn, The embedding of firs in skew fields, Proc. London Math. Soc. (3) 23 (1971), 193–213. MR 297814, DOI 10.1112/plms/s3-23.2.193
Jacques Lewin and Tekla Lewin, An embedding of the group algebra of a torsion-free one-relator group in a field, J. Algebra 52 (1978), no. 1, 39–74. MR 485972, DOI 10.1016/0021-8693(78)90260-0
P. M. Cohn, Free rings and their relations, London Mathematical Society Monographs, No. 2, Academic Press, London-New York, 1971. MR 0371938
Peter Malcolmson, A prime matrix ideal yields a skew field, J. London Math. Soc. (2) 18 (1978), no. 2, 221–233. MR 509937, DOI 10.1112/jlms/s2-18.2.221
A. I. Lichtman, On normal subgroups of multiplicative group of skew fields generated by a polycyclic-by-finite group, J. Algebra 78 (1982), no. 2, 548–577. MR 680374, DOI 10.1016/0021-8693(82)90095-3
A. I. Lichtman, On linear groups over a field of fractions of a polycyclic group ring, Israel J. Math. 42 (1982), no. 4, 318–326. MR 682316, DOI 10.1007/BF02761413
A. I. Lichtman, Matrix rings and linear groups over a field of fractions of enveloping algebras and group rings. I, J. Algebra 88 (1984), no. 1, 1–37. MR 741929, DOI 10.1016/0021-8693(84)90087-5
A. I. Lichtman, On matrix rings and linear groups over fields of fractions of group rings and enveloping algebras. II, J. Algebra 90 (1984), no. 2, 516–527. MR 760026, DOI 10.1016/0021-8693(84)90187-X
A. I. Lichtman, Free subgroups in linear groups over some skew fields, J. Algebra 105 (1987), no. 1, 1–28. MR 871744, DOI 10.1016/0021-8693(87)90177-3
Donald S. Passman, The algebraic structure of group rings, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1977. MR 0470211
A. I. Mal′cev, Generalized nilpotent algebras and their associated groups, Mat. Sbornik N.S. 25(67) (1949), 347–366 (Russian). MR 0032644

The theory of groups