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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
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
Summary:
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 $S$.
References:
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