A note on Skolem-Noether algebras

Juncheol Han; Tsiu-Kwen Lee; Sangwon Park

Czechoslovak Mathematical Journal (2021)

  • Issue: 1, page 137-154
  • ISSN: 0011-4642

Abstract

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The paper was motivated by Kovacs’ paper (1973), Isaacs’ paper (1980) and a recent paper, due to Brešar 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 : M n ( K ) → M n ( S ) be extended to an inner automorphism of M n ( 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 .

How to cite

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Han, Juncheol, Lee, Tsiu-Kwen, and Park, Sangwon. "A note on Skolem-Noether algebras." Czechoslovak Mathematical Journal (2021): 137-154. <http://eudml.org/doc/297306>.

@article{Han2021,
abstract = {The paper was motivated by Kovacs’ paper (1973), Isaacs’ paper (1980) and a recent paper, due to Brešar 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 $S$, $n\in \mathbb \{N\}$, the goal of this paper is to study the question: when can a homomorphism $\phi \colon \{\rm M\}_n(K)\rightarrow \{\rm M\}_n(S)$ be extended to an inner automorphism of $\{\rm M\}_n(S)$? 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 $S$.},
author = {Han, Juncheol, Lee, Tsiu-Kwen, Park, Sangwon},
journal = {Czechoslovak Mathematical Journal},
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},
language = {eng},
number = {1},
pages = {137-154},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A note on Skolem-Noether algebras},
url = {http://eudml.org/doc/297306},
year = {2021},
}

TY - JOUR
AU - Han, Juncheol
AU - Lee, Tsiu-Kwen
AU - Park, Sangwon
TI - A note on Skolem-Noether algebras
JO - Czechoslovak Mathematical Journal
PY - 2021
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
IS - 1
SP - 137
EP - 154
AB - The paper was motivated by Kovacs’ paper (1973), Isaacs’ paper (1980) and a recent paper, due to Brešar 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 $S$, $n\in \mathbb {N}$, the goal of this paper is to study the question: when can a homomorphism $\phi \colon {\rm M}_n(K)\rightarrow {\rm M}_n(S)$ be extended to an inner automorphism of ${\rm M}_n(S)$? 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 $S$.
LA - eng
KW - 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
UR - http://eudml.org/doc/297306
ER -

References

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  7. Lam, T. Y., 10.1007/978-1-4684-0406-7, Graduate Texts in Mathematics 131, Springer, New York (1991). (1991) Zbl0728.16001MR1125071DOI10.1007/978-1-4684-0406-7
  8. McCoy, N. H., 10.1215/S0012-7094-45-01232-4, Duke Math. J. 12 (1945), 381-387. (1945) Zbl0060.05901MR0012266DOI10.1215/S0012-7094-45-01232-4
  9. Milinski, A., 10.1080/00927879308824760, Commun. Algebra 21 (1993), 3719-3725. (1993) Zbl0793.16030MR1231628DOI10.1080/00927879308824760
  10. Rosenberg, A., Zelinsky, D., 10.2140/pjm.1961.11.1109, Pac. J. Math. 11 (1961), 1109-1117. (1961) Zbl0116.02501MR0148709DOI10.2140/pjm.1961.11.1109
  11. Rowen, L., 10.1090/S0002-9904-1973-13162-3, Bull. Am. Math. Soc. 79 (1973), 219-223. (1973) Zbl0252.16007MR0309996DOI10.1090/S0002-9904-1973-13162-3
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