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.
group algebra, invariant ideal, locally finite simple group, finitary permutation group, permutation module, finitary linear group
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