Let $A=C(0,1)$ be the ring of continuous real valued functions on the **open** interval $(0,1)$. It is not too difficult to show that if $\mathfrak{m}\subseteq A$ is a maximal ideal with residue field $A/\mathfrak{m}\simeq \mathbb{R}$ then $\mathfrak{m}=\mathfrak{m}_a:=ker(ev_a)$ where for $a\in(0,1)$,
$ev_a:A\rightarrow\mathbb{R}$ is the evaluation map at $a$. It is easy to show that there are
maximal ideals not of the for $\mathfrak{m}_a$. For instance one may look at the
ideal

$$ I=\{f\in A:\exists n\in\mathbf{Z}_{\geq 1}, f\equiv 0\;\;\mbox{on $(0,1/n]$}\} $$ Then $I$ is not contained in any $\mathfrak{m}_a$ but by Zorn's lemma it is contained in some maximal ideal $\mathfrak{M}$. So here are two natural questions on the ring $A$ for which I don't have an answer:

Q1: Do we have a structure theorem for the possible residue fields of maximal ideals of $A$.

Q2: How does one show the existence (or construct) of a prime ideal of $A$ which is not maximal?