I should recall the notion of maximal subring of a commutative unitary ring $R$.

Def:A commutative ring $S$ is called a maximal subring of $R$ if $S \subset R$ and if $T \subset R$ constitute a commutative ring with the restricted addition and multiplication of $R$ and also $S\subsetneq T$ then we could deduce that $T=R$.

I am interested in studing this notion in **Rings of continuous functions**.

We could easily deduce that for $x \neq y \in X$ The set of the form $$M_{x,y}=\Big(f\in C(X): f(x)=f(y) \Big)$$ forms a maximal subring of $C(X)$

From the above summary and notations I could pose my Questions.

Question1:Is there a maximal subring in $C(X)$ other than all $M_{x,y}$'s?

Question2:Is $X$ compact if all maximal subrings of $C(X)$ is of the form $M_{x,y}$?

PS:I suppose that all subrings of a commutative ring $R$ contains the unitary element of $R$.