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 \omega K[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.
PASSMAN, D. S. (2007) "Finitary Actions and Invariant Ideals," Turkish Journal of Mathematics: Vol. 31: No. 5, Article 9. Available at: https://journals.tubitak.gov.tr/math/vol31/iss5/9