Summary: THE CONTROLLER SUBGROUP OF ONE­SIDED IDEALS IN
COMPLETED GROUP RINGS
KONSTANTIN ARDAKOV
Abstract. Let G be a compact p­adic analytic group and let I be a right
ideal of the Iwasawa algebra kG. A closed subgroup H of G is said to control
I if I can be generated as a right ideal by a subset of kH. We prove that the
intersection of any collection of such subgroups again controls I. This has an
application to the study of two­sided ideals in nilpotent Iwasawa algebras.
1. Introduction
1.1. Controlling subgroups. Let G be a group and let k be a field. A subgroup
H of G is said to control a right ideal I of the group algebra k[G] if I can be
generated as a right ideal by a subset of the subalgebra k[H ] of k[G], or equivalently,
if I = (I # k[H]) · k[G]. It is clear that if I is controlled by a proper subgroup H
then I is completely determined by a right ideal in a smaller group algebra, namely
I # k[H ]. In the study of two­sided ideals in group rings, theorems that assert
that under suitable conditions a two­sided ideal is controlled by a known small
subgroup of the group are particularly desirable: a canonical example of such a
result is Zalesskii's Theorem [11], which asserts that every faithful prime ideal of
the group algebra of a finitely generated torsion­free nilpotent group is controlled
by the centre of the group.

Source: Ardakov, Konstantin - School of Mathematical Sciences, University of Nottingham

Collections: Mathematics