Primeness, semiprimeness and localisation in Iwasawa algebras

TL;DR

This paper establishes criteria for the primeness, semiprimeness, and domain properties of completed group algebras of p-adic analytic groups, extending previous algebraic results and analyzing their localizations and dimensions.

Contribution

It provides necessary and sufficient conditions for algebraic properties of Iwasawa algebras and explores their localizations and homological dimensions, advancing the understanding of their structure.

Findings

Criteria for primeness and semiprimeness of Iwasawa algebras.

Conditions for localization at semiprime ideals.

Results on Krull and global dimensions of localizations.

Abstract

Necessary and sufficient conditions are given for the completed group algebras of a compact p-adic analytic group with coefficient ring the p-adic integers or the field of p elements to be prime, semiprime and a domain. Necessary and sufficient conditions for the localisation at semiprime ideals related to the augmentation ideals of closed normal subgroups are found. Some information is obtained about the Krull and global dimensions of the localisations. The results extend and complete work of A. Neumann and J. Coates et al.