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Let $S=M_n(D)$, the ring of $n\times n$ matrices with entries in a division ring $D$. Now suppose that $R$ is a simple unital Artinian subring of $S$. Is it the case that $R\cong M_k(D')$ for some positive integer $k\leq n$ and subdivision ring $D'$ of $D$? To be clear, I am aware of the Wedderburn-Artin Theorem, so I know $R$ is isomorphic to some matrix ring over some division ring. But I'm asking if we can extract a bit more information by choosing the division ring to be a subdivision ring of $D$.

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This is not true, and there is a well-known counter-example. Take, for example, the embedding of the Hamilton quaternions into $M_4(\mathbb{R})$.

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  • $\begingroup$ ah yes, of course. Thank you! $\endgroup$
    – Greg
    May 19, 2022 at 2:25

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