Given a division ring $R$. A field $F$ is contained in $R$. Show that $F \subseteq Z(R)$.
I think this statement is correct, but I find it difficult for me to prove it. I am aware of the fact that for a division ring $R$, the center of $R$, $Z(R)$ is a field.
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 $Z(\mathbb{H})=\mathbb{R}\not\supseteq\mathbb{C}$.
