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Suppose that $A$ is an infinite definable subset of a real closed field $R$ which is a zero-divisor-free ring under operations whose graphs are definable in $R$. Then $A$ is definably isomorphic to one of $R$, $R(\sqrt{-1})$ or the ring of quaternions over $R$. This is a special case of the main result of:

Otero, Peterzil, and Pillay, On groups and rings definable in o-minimal expansions of real closed fields. (English summary) , Bull. London Math. Soc. 28 (1996), no. 1, 7–14.

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Suppose that $A$ is an infinite definable subset of a real closed field $R$ which is a zero-divisor-free ring under operations whose graphs are definable in $R$. Then $A$ is definably isomorphic to one of $R$, $R(\sqrt{-1})$ or the ring of quaternions over $R$. This is a special case of the main result of:

Otero, Peterzil, and Pillay, On groups and rings definable in o-minimal expansions of real closed fields. (English summary) Bull. London Math. Soc. 28 (1996), no. 1, 7–14.
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