Let K[H] denote the group algebra of an infinite locally finite group
H. In recent years, the lattice of ideals of K[H] has been
extensively studied under the assumption that H is simple.
From these many results, it appears that such group algebras tend to
have very few ideals. While some work still remains to be done in the
simple group case, we nevertheless move on to the next stage of this
program by considering certain abelian-by-(quasi-simple) groups.
Standard arguments reduce this problem to that of characterizing the
ideals of an abelian group algebra K[V] stable under the action of an
appropriate automorphism group of V. Specifically, in this paper, we
let G be a quasi-simple group of Lie type defined over an infinite
locally finite field F, and we let V be a finite-dimensional vector
space over a field E of the same characteristic p. If G acts
nontrivially on V by way of the homomorphism
G -> GL(V), and if V has no proper G-stable subgroups, then we show that the
augmentation ideal wK[V] is the unique proper G-stable ideal of
K[V] when char K is different from p. The proof of this result requires, among
other things, that we study characteristic p division rings D, certain
multiplicative subgroups M of D, and the action of M on the
group algebra K[A], where A is the
additive group of D. In particular, properties of the quasi-simple
group G come into play only in the last section of this paper.
We remark that the final value of a polynomial form f: A -> S need not
be a subgroup of S. See "Polynomial and inverse forms" below for a counterexample.