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Skolem–Noether algebras
Journal article   Peer reviewed

Skolem–Noether algebras

Matej Brešar, Christoph Hanselka, Igor Klep and Jurij Volčič
Journal of algebra, v 498, pp 294-314
15 Mar 2018
DOI: https://doi.org/10.1016/j.jalgebra.2017.11.045
url
https://doi.org/10.1016/j.jalgebra.2017.11.045View
Published, Version of Record (VoR) Restricted

Abstract

Artinian algebra Automorphism of a tensor product Central simple algebra Inner automorphism Semilocal ring Skolem–Noether theorem Sylvester domain
An algebra S is called a Skolem–Noether algebra (SN algebra for short) if for every central simple algebra R, every homomorphism R→R⊗S extends to an inner automorphism of R⊗S. One of the important properties of such an algebra is that each automorphism of a matrix algebra over S is the composition of an inner automorphism with an automorphism of S. The bulk of the paper is devoted to finding properties and examples of SN algebras. The classical Skolem–Noether theorem implies that every central simple algebra is SN. In this article it is shown that actually so is every semilocal, and hence every finite-dimensional algebra. Not every domain is SN, but, for instance, unique factorization domains, polynomial algebras and free algebras are. Further, an algebra S is SN if and only if the power series algebra S[[ξ]] is SN.

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