Invariant ideals of abelian group algebras under the torus action of a field, II

Let V = V(1) circle plus V(2) be a finite-dimensional vector space over an infinite locally-finite field F. Then V admits the torus action of G = F degrees by defining (v(1) circle plus v(2))(g) = v(1)g(-1) circle plus v(2)g. If K is a field of characteristic different from that of F, then G acts on the group algebra K[V] and it is an interesting problem to determine all G-stable ideals of this algebra. In this paper, we show that, for almost all fields F. the G-stable ideals are uniquely writable as finite irredundant intersections of augmentation ideals of subspaces W(1) circle plus W(2), with W(1) subset of V(1) and W(2) subset of V(2). As a consequence, the set of all G-stable ideals is Noetherian. (C) 2011 Elsevier Inc. All rights reserved.