A Note on Skolem-Noether Algebras

Abstract

The paper was motivated by Kovacs’ paper (1973), Isaacs’ paper (1980) and a recent paper, due to Bresar et al. (2018), concerning Skolem-Noether algebras. Let K be a unital commutative ring, not necessarily a field. Given a unital K-algebra S, where K is contained in the center of S, n ∈ ℕ, the goal of this paper is to study the question: when can a homomorphism ϕ: Mn(K) → Mn(S) be extended to an inner automorphism of Mn(S)? As an application of main results presented in the paper, it is proved that if S is a semilocal algebra with a central separable subalgebra R, then any homomorphism from R into S can be extended to an inner automorphism of S.

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Acknowledgments

The authors would like to express their thanks to the referee for valuable suggestions and careful reading.

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Correspondence to Tsiu-Kwen Lee.

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The work of T.-K. Lee was supported in part by the Ministry of Science and Technology of Taiwan (MOST 107-2115-M-002-018-MY2) and the NCTS (Taipei Office).

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Han, J., Lee, TK. & Park, S. A Note on Skolem-Noether Algebras. Czech Math J 71, 137–154 (2021). https://doi.org/10.21136/CMJ.2020.0215-19

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Keywords

  • Skolem-Noether algebra
  • (inner) automorphism
  • matrix algebra
  • central simple algebra
  • central separable algebra
  • semilocal ring
  • unique factorization domain (UFD)
  • stably finite ring
  • Dedekind-finite ring

MSC 2020

  • 16K20
  • 16W20
  • 16S50