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Maximal prime homomorphic images of mod-p Iwasawa algebras

Part of: Modules, bimodules and ideals Rings and algebras arising under various constructions

Published online by Cambridge University Press:  05 March 2021

WILLIAM WOODS
WILLIAM WOODS*
Affiliation:
Pathways Department, University of Essex, Colchester, CO4 3SQ, e-mail: [email protected]
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Abstract

Let k be a finite field of characteristic p, and G a compact p-adic analytic group. Write kG for the completed group ring of G over k. In this paper, we describe the structure of the ring kG/P, where P is a minimal prime ideal of kG. We give an explicit isomorphism between kG/P and a matrix ring with coefficients in the ring ${(k'G')_\alpha }$ , where $k'/k$ is a finite field extension, $G'$ is a large subquotient of G with no finite normal subgroups, and (–)α is a “twisting” operation that preserves many desirable properties of the ring structure. We demonstrate the usefulness of this isomorphism by studying the correspondence induced between certain ideals of kG and those of ${(k'G')_\alpha }$ , and showing that this preserves many useful “group-theoretic” properties of ideals, in particular almost-faithfulness and control by a closed normal subgroup.

MSC classification

Primary: 16S34: Group rings , Laurent polynomial rings 16S35: Twisted and skew group rings, crossed products 16D25: Ideals
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 171 , Issue 2 , September 2021 , pp. 387 - 419
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

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