Finitary Actions and Invariant Ideals

Let K be a field and let G be a group. If G acts on an abelian group V , then it acts naturally on any group algebra K[V ], and we are concerned with classifying the G-stable ideals of K[V ]. In this paper, we consider a rather concrete situation. We take G to be an infinite locally finite simple group acting in a finitary manner on V . When G is a finitary version of a classical linear group, then we show that the augmentation ideal ωK[G] is the unique proper G-stable ideal of K[V ]. On the other hand, if G is a finitary alternating group acting on a suitable permutation module V , then there is a rich family of G-stable ideals of K[V ], and we show that these behave like certain graded ideals in a polynomial ring.