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

Let $D$ be a division ring and let $V=D^n$ be a finite-dimensional $D$-vector space, viewed multiplicatively. If $G=D^\bullet$is the multiplicative group of $D$, then $G$ acts on $V$ and hence on any group algebra $K[V]$. Our goal is to completely describe the semiprime $G$-stable ideals of $K[V]$. 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.