Field contained in a division ring lies in the center of the ring

The quaternions $\Bbb H=\{a+bi+cj+dk:a,b,c,d\in\Bbb R\}$ contains the subring $\Bbb C=\{a+bi:a,b,\in \Bbb R\}$ but $\Bbb C$ is not contained in the centre of $\Bbb H$ which is $\Bbb R$.


This is false. For instance, the quaternions $\mathbb{H}$ are a division ring and contain $\mathbb{C}$ as a subfield but the $Z(\mathbb{H})=\mathbb{R}\not\supseteq\mathbb{C}$.

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Abstract Algebra

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