Abstract:Let KG be the group ring of a polycyclic by finite group G over a field K of characteristic zero. It is proved that if $e = \sum e(g)g$ is a nontrivial idempotent in KG then its trace $e(1)$ is a rational number $r/s,(r,s) = 1$, with the property that for every prime divisor p of s, G has an element of order p. This result is used to prove that if R is a commutative ring of characteristic zero, without nontrivial idempotents and G is a polycyclic by finite group such that no group order $\ne 1$ is invertible in R, then RG has no nontrivial idempotents.
- Akira Hattori, Rank element of a projective module, Nagoya Math. J. 25 (1965), 113–120. MR 175950
- Edward Formanek, Idempotents in Noetherian group rings, Canadian J. Math. 25 (1973), 366–369. MR 316494, DOI 10.4153/CJM-1973-037-6
- M. M. Parmenter and S. K. Sehgal, Idempotent elements and ideals in group rings and the intersection theorem, Arch. Math. (Basel) 24 (1973), 586–600. MR 335569, DOI 10.1007/BF01228258
- Donald S. Passman, Infinite group rings, Pure and Applied Mathematics, vol. 6, Marcel Dekker, Inc., New York, 1971. MR 0314951
- Sudarshan K. Sehgal, Certain algebraic elements in group rings, Arch. Math. (Basel) 26 (1975), 139–143. MR 369417, DOI 10.1007/BF01229717 S. Sehgal and H. Zassenhaus, Group rings without non-trivial idempotents, Arch. Math. (to appear).
- A. E. Zalesskiĭ, A certain conjecture of Kaplansky, Dokl. Akad. Nauk SSSR 203 (1972), 749–751 (Russian). MR 0297895
- © Copyright 1977 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 62 (1977), 11-14
- MSC: Primary 16A26
- DOI: https://doi.org/10.1090/S0002-9939-1977-0427361-9
- MathSciNet review: 0427361