TITLE: Prime ideals in $C[0,1]$
QUESTION [14 upvotes]: Are there any prime ideals in the ring $C[0,1]$ of continuous functions $[0,1]\rightarrow \mathbb{R}$, which are not maximal?
Perhaps, I duplicate smb's question, but this is an interesting problem!
Could you give me any hint or give a link to some literature?

REPLY [3 votes]: Fix a point $c\in (0, 1)$. Define $I := \{ f \in C[0,1] : Z(f)$ contains an open nbd of $c \}$ where $Z(f)$ denotes the zero set of $f$. It can be easily checked that $I$ is a radical ideal and $I \subsetneq M_c := \{ f \in C[0,1] : f(c) = 0 \}$. Also note that $I \nsubseteq M_b$ for any $b \in [0, 1], b\neq c$. Now use the following fact: In a commutative ring with unity, a radical ideal is an intersection of prime ideals. So, $I$ is an intersection of prime ideals, but clearly it's not an intersection of maximal ideals only. Thus there must exists at least one non-maximal prime.