Skolem–Noether algebras

Matej Brešar, Christoph Hanselka, Igor Klep, Jurij Volčič

Research output: Contribution to journalArticlepeer-review

2 Scopus citations


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.

Original languageEnglish
Pages (from-to)294-314
Number of pages21
JournalJournal of Algebra
StatePublished - 15 Mar 2018
Externally publishedYes


  • Artinian algebra
  • Automorphism of a tensor product
  • Central simple algebra
  • Inner automorphism
  • Semilocal ring
  • Skolem–Noether theorem
  • Sylvester domain

ASJC Scopus subject areas

  • Algebra and Number Theory


Dive into the research topics of 'Skolem–Noether algebras'. Together they form a unique fingerprint.

Cite this