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A note on Skolem-Noether algebras
Czechoslovak Mathematical Journal, Tome 71 (2021) no. 1, pp. 137-154
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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 $S$, $n\in \mathbb N$, the goal of this paper is to study the question: when can a homomorphism $\phi \colon {\rm M}_n(K)\to {\rm M}_n(S)$ 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 $S$.
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 $\phi \colon {\rm M}_n(K)\to {\rm M}_n(S)$ 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 $S$.
DOI : 10.21136/CMJ.2020.0215-19
Classification : 16K20, 16S50, 16W20
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 
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Han, Juncheol; Lee, Tsiu-Kwen; Park, Sangwon. A note on Skolem-Noether algebras. Czechoslovak Mathematical Journal, Tome 71 (2021) no. 1, pp. 137-154. doi: 10.21136/CMJ.2020.0215-19

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