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

Abstract

Let D D be a division ring and let V = D n V=D^n be a finite-dimensional right 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] . In this paper, we completely describe the semiprime G G -stable ideals of K [ V ] K[V] , and conclude that these ideals satisfy the ascending chain condition. 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 (handled in Part I).