Abstract
The main theorem gives necessary and sufficient conditions for the rational group algebra QG to be without (nonzero) nilpotent elements if G is a nilpotent or F·C group. For finite groups G, a characterisation of group rings RG over a commutative ring with the same property is given. As an application those nilpotent or F·C groups are characterised which have the group of units U(KG) solvable for certain fields K.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
AYOUB, R.G., AYOUB, C.: On the group ring of a finite abelian group. Bull. Austral. Math. Soc.1, 245–261 (1969).
BATEMAN, J.M.: On the solvability of unit groups of group algebras. Trans. Amer. Math. Soc.157, 73–86 (1971).
BERMAN, S.D.: On certain properties of integral group rings. Dokl. Akad. Nauk SSSR (N.S.)91 7–9 (1953).
BOVDI, A.A., MIHOVSKI, S.V.: Idempotents in crossed products. Soviet Math. Dokl.11, 1439–1441 (1971).
FISHER, J.L., PARMENTER, M.M., SEHGAL, S.K.: Group rings with solvable n-Engel unit groups (to appear).
LANSKI, C.: Some Remarks On Rings With Solvable Units, Ring Theory. New York: Academic Press 1972.
MOSER, C.: Representation de -1 comme somme de carres dans un corps cyclotomique quelconque. J. Number Theory5, 139–141 (1973).
PASSMAN, D.S.: Advances in group rings. Israel Math. J. (to appear).
PASSMAN, D.S.: Infinite Group Rings. New York: Marcel Dekker 1971.
ROBINSON, D.J.S.: Finiteness Conditions And Generalized Solvable Groups. New York: Springer-Verlag 1972.
Author information
Authors and Affiliations
Additional information
This work has been supported by N.R.C. Grant No. A-5300.
Rights and permissions
About this article
Cite this article
Sehgal, S.K. Nilpotent elements in group rings. Manuscripta Math 15, 65–80 (1975). https://doi.org/10.1007/BF01168879
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01168879