# Skolem-Noether algebras

## TL;DR

This paper investigates Skolem-Noether algebras (SN algebras), characterizing their properties, providing examples, and establishing that many classes of algebras, including semilocal and polynomial algebras, are SN.

## Contribution

It extends the classical Skolem-Noether theorem to broader classes of algebras and explores conditions under which an algebra is SN, including polynomial and free algebras.

## Key findings

- Every semilocal and finite-dimensional algebra is SN.
- UFDs, polynomial, and free algebras are SN.
- SN property is preserved under power series extension.

## Abstract

An algebra $S$ is called a Skolem-Noether algebra (SN algebra for short) if for every central simple algebra $R$, every homomorphism $R\to R\otimes S$ extends to an inner automorphism of $R\otimes S$. 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 $S[[\xi]]$ is SN.