Examples of intersection of maximal ideal with a subring not being maximal in the subring.

Question

Let $R$ be a commutative ring with identity, $R_0 \subset R$ be a subring, and $m \triangleleft R$ be a maximal ideal. I have already shown that $m \cap R_0 \triangleleft R_0$ must be a prime ideal, and it seems to me that it can't be maximal, but I can't find/think of any good examples for this.

Answer 1

Suppose that $R$ is a field, $m=0$, and $R_0 \subset R$ is any proper subring.

Then $m_0 = m\cap R_0 = 0$, so the only way $m_0$ can be maximal is if $R_0$ is a field.

But there are lots of subrings of fields that aren't fields. In fact, any integral domain can be embedded into its field of fractions. For a simple example, take $\mathbb{Z}\subset\mathbb{Q}$.