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Skolem-Noether algebras. (English) Zbl 1439.16016
Summary: An algebra \(S\) is called a Skolem-Noether algebra (SN algebra for short) if for every central simple algebra \(R\), every homomorphism \(R \rightarrow R \otimes S\) 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.

MSC:
16K20 Finite-dimensional division rings
16W20 Automorphisms and endomorphisms
16L30 Noncommutative local and semilocal rings, perfect rings
16U10 Integral domains (associative rings and algebras)
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