TITLE: What are the maximal ideals in $C(0,1)$ apart from $M_c = \{f\in C(0,1) | f(c)=0\}$?
QUESTION [6 upvotes]: I have studied about maximal ideals in $C[0,1]$ . They are precisely of the form $$\{f\in C[0,1] | f(c)=0\} \text{ for } c\in [0,1].$$
If we replace $[0,1]$ by $(0,1)$ and then look at $C(0,1)$, then obviously  $M_c = \{f\in C(0,1) | f(c)=0\}$ are maximal ideals in $C(0,1)$ , $\forall c \in (0,1)$ , but since we do not have the compactness of $[0,1]$ anymore, I guess there are some other maximal ideals as well. 
I am trying to solve the problem by first giving an existential argument and then by showing an explicit maximal ideal other that of the form $M_c$ for some $c \in (0,1)$ .

REPLY [5 votes]: Since $\mathbb{R}$ and $(0,1)$ are homeomorphic, you can look at maximal ideal in $C(\mathbb R)$.
There is the idea form Dummit and Foote Abstract Algebra, page 259 Ex 34.
Let $I\subset C(\mathbb R)$ be the set of all continuous functions with compact support. It can be shown that $I$ is an ideal. Let $M$ be a maximal ideal containing $I$. Then $M\neq M_{c}$ for any $c\in \mathbb R$. For suppose $M=M_{c}$, and let $r=|c|+1$. There is a function $f\in C(\mathbb R)$, such that $f(x)>0$ when $x\in (-|c|,|c|)$ and $f(x)=0$ when $x>r$ or $x<-r$. Then $f\in I$ but $f\notin M_{c}$.