Invariant ideals of abelian group algebras under the multiplicative action of a field. I

Abstract

Let D D be a division ring and let V = D n V=D^n be a finite-dimensional D D -vector space, viewed multiplicatively. If G = D ∙ G=D^\bullet is the multiplicative group of D D , then G G acts on V V and hence on any group algebra K [ V ] K[V] . Our goal is to completely describe the semiprime G G -stable ideals of K [ V ] K[V] . As it turns out, this result follows fairly easily from the corresponding results for the field of rational numbers (due to Brookes and Evans) and for infinite locally-finite fields. Part I of this work is concerned with the latter situation, while Part II deals with arbitrary division rings.