Prime ideals in ring of continuous functions

Question

Let $R:=C(\mathbb R,\mathbb R)$ be the ring of continuous $\mathbb R$-valued functions on $\mathbb R$. Is every prime ideal in $R$ maximal?

Answer 1

No, because that would imply $J(R)$ is nil and $R/J(R)$ is von Neumann regular. (I am denoting the Jacobson radical of $R$ as $J(R)$.)

But it is easy to show in this ring that $J(R)$ is $\{0\}$ and $R$ is not von Neumann regular. (In fact, this ring has only trivial idempotents, but there must be many idempotents in a von Neumann regular ring.)

You can read elsewhere about the connection of prime ideals in this ring to ultrafilters. This is pretty much the best description you can get for prime ideals in such a ring, and it shows that in general prime ideals in a ring like this do not have to be maximal.