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# Maximalfunctional subringsof$C(X)$

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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 >$T$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$R$. 2 added 2 characters in body; added 3 characters in body 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$T >$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 $R$.

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