Is there a local ring $O$ such that $\mathbb R\subsetneq O$ and $\mathbb C\nsubseteq O$?

Question

Is there a local ring $O$ such that $\mathbb R\subsetneq O$ and $\mathbb C\nsubseteq O$?

I need this to prove a another problem I'm proving.

I hope there aren't such a ring.

Thanks in advance

Answer 1

Collecting the comments together: in any UFD (we can probably relax this a bit but I don't feel like thinking about it) $D$, if $\pi\in D$ is irreducible then $D/(\pi^2)$ is local with unique maximal ideal $(\pi)$, and the localization of $D$ at $\pi$ also has unique maximal ideal $(\pi)$. Given any field $F$, we can form the polynomial ring $F[x]$, which is a UFD. If $\pi\in F[x]$ is irreducible, then $F[x]/(\pi^2)$ has no proper subring containing $F$ strictly (exercise - hint: all elements look like $a+b\pi$), and since this ring is not a domain it isn't a field so contains no field extension of $F$. And for the other example, the localization of $F[x]$ at $\pi$ is contained in its fraction field $F(x)$ which contains no algebraic extensions of $F$ (since every element outside $F$ is transcendental over $F$).

All of these general facts inform the examples $\Bbb R[T]/(T^2)$ and $\Bbb R[T]_{(T)}$.