Prime ideals in nilpotent Iwasawa algebras

TL;DR

This paper investigates the structure of prime ideals in Iwasawa algebras of nilpotent p-valued groups, revealing they are controlled by the group's center and that all prime ideals are completely prime, with a focus on non-commutative valuations.

Contribution

It establishes that faithful prime ideals are controlled by the center of the group and that the prime spectrum decomposes into commutative strata, introducing a non-commutative valuation technique.

Findings

Prime ideals are controlled by the group's center.

The prime spectrum decomposes into disjoint commutative strata.

All prime ideals are completely prime.

Abstract

Let G be a nilpotent complete p-valued group of finite rank and let k be a field of characteristic p. We prove that every faithful prime ideal of the Iwasawa algebra kG is controlled by the centre of G, and use this to show that the prime spectrum of kG is a disjoint union of commutative strata. We also show that every prime ideal of kG is completely prime. The key ingredient in the proof is the construction of a non-commutative valuation on certain filtered simple Artinian rings.