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

Let be a division ring and let be a finite-dimensional -vector space, viewed multiplicatively. If is the multiplicative group of , then acts on and hence on any group algebra . Our goal is to completely describe the semiprime -stable ideals of . 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.