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Extensions of valuation rings in central simple algebras


Authors: H.-H. Brungs and J. Gräter
Journal: Trans. Amer. Math. Soc. 317 (1990), 287-302
MSC: Primary 16A39
DOI: https://doi.org/10.1090/S0002-9947-1990-0946216-5
MathSciNet review: 946216
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Abstract: Certain subrings $ R$ of simple algebras $ Q$, finite dimensional over their center $ K$, are studied. These rings are called $ Q$-valuation rings since they share many properties with commutative valuation rings. Let $ V$ be a valuation ring of $ K$, the center of $ Q$, and let $ \mathcal{R}$ be the set of $ Q$-valuation rings $ R$ in $ Q$ with $ R \cap K = V$, then $ \left\vert \mathcal{R} \right\vert \geq 1$. This extension theorem, which does not hold if one considers only total valuation rings, was proved by N. I. Dubrovin. Here, first a somewhat different proof of this result is given and then information about the set $ \mathcal{R}$ is obtained. Theorem. The elements in $ \mathcal{R}$ are conjugate if $ V$ has finite rank. Theorem. The elements in $ \mathcal{R}$ are total valuation rings if $ \mathcal{R}$ contains one total valuation ring. In this case $ Q$ is a division ring. Theorem. $ \mathcal{R}$ if $ \mathcal{R}$ contains an invariant total valuation ring.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1990-0946216-5
Keywords: Valuation ring, simple artinian algebra, division ring, extensions, completions, Galois theory, localization, Bezout order
Article copyright: © Copyright 1990 American Mathematical Society

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