A-INVARIANT IDEALS OF CROSSED PRODUCTS

Abstract

Let K * G be a crossed product of the group G over the F-algebra K and let F be a G-invariant subfield of K with characteristic p ≥ 0. Then G(F) = {g ∈ G | α ^gσ = α for all α ∈ F} is a normal subgroup of G and I = (I ∩ K * G(F)) K * G for every ideal I of K * G. Let H be a normal subgroup of G(F) such that G(F)/H is a solvable group. If all factors of the commutator series of G(F)/H have no p-elements when p > 0, then J(K * G) ⊆ J (K * H) K * G, where J (K * G) is the radical of Jacobson. Moreover, if K is a semiprimitive F-algebra and G(F) is a locally nilpotent group without p-elements in case p > 0, then K * G is semiprimitive.

ACKNOWLEDGMENTS

The work was supported by the fund NIMP of the Plovdiv University under contract PU-9-MM/99.