Skolem-Noether algebras

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.