Abstract
In this paper we obtain necessary and sufficient conditions for the crossed productR *G to be prime or semiprime under the assumption thatR is prime. The main techniques used are the Δ-methods which reduce these questions to the finite normal subgroups ofG and a study of theX-inner automorphisms ofR which enables us to handle these finite groups. In particular we show thatR *G is semiprime ifR has characteristic 0. Furthermore, ifR has characteristicp>0, thenR *G is semiprime if and only ifR *P is semiprime for all elementary abelianp-subgroupsP of Δ+(G) ∩G inn.
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,On rings of quotients, Symposia Mathematica8 (1972), 149–164.
J. W. Fisher and S. Montgomery,Semi-prime skew group rings, J. Algebra.
V. K. Kharchenko,Generalized identities with automorphisms, Algebra and Logic14 (1975), 132–148.
W. S. Martindale,Prime rings satisfying a generalized polynomial identity, J. Algebra12 (1969), 576–584.
D. S. Passman,Radicals of twisted group rings, Proc. London Math. Soc. (3)20 (1970), 409–437.
D. S. Passman,The Algebraic Structure of Group Rings, Wiley-Interscience, New York, 1977.
A. Reid,Semi-prime twisted group rings, J. London Math. Soc. (2)12 (1976), 413–418.
M. K. Smith,Group algebras, J. Algebra18 (1971), 477–499.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Montgomery, S., Passman, D.S. Crossed products over prime rings. Israel J. Math. 31, 224–256 (1978). https://doi.org/10.1007/BF02761494
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02761494