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Let $C[0,1]$ be the ring of continuous real-valued functions on $[0,1]$, with addition and multiplication defined pointwise. For any subset $S$ of $C[0,1]$ let $Z(S)=\{f\in C[0,1]: f(x)=0 \text{ for all }x\in S\}$. Then which of the following statements are true?
(a) If $Z(S)$ is an ideal in $C[0,1]$ then $S$ is closed in $[0,1]$.
(b) If $Z(S)$ is a maximal ideal then $S$ has only one point.
(c) If $S$ has only one point then $Z(S)$ is a maximal ideal.

(a) not necessary, If I take $S=(1/2,1/3)$ still $Z(S)$ is an ideal.

(b) I know that maximal ideals in $C[0,1]$ comes in this fashion (I don't know the proof rigorously) i.e $C_a=\{f\in C[0,1]:f(a)=0\}$ so $S$ may be finite or countable set? So I guess $b$ is true statement and for the same reason $c$ is also true. But I will be happy if some one explain me a bit about (b) and (c). Thank you.

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Could you provide us with a reference? – Hans Giebenrath Dec 18 '12 at 9:06
it is not from a book just a question paper of past year exam paper – La Belle Noiseuse Dec 18 '12 at 9:12

1 Answer 1

To show that $C_a$ is maximal define $\phi:C[0,1]\longrightarrow\mathbb R$ with $\phi(f)=f(a)$ and then use the first isomorphism theorem.
For the converse define $\phi:C[0,1]\longrightarrow\mathbb C(S), \ f \mapsto f|_S$ to show that $C[0,1]\ /Z(S)$ is isomorphic to $C(S)$.

Then use that $m \leq R$ is maximal $\iff R/m$ is a field.

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$S$ can be uncountable or countably infinite to be a maximal ideal? – La Belle Noiseuse Dec 18 '12 at 7:03
@Kuttus: $Z(S)$ is maximal iff $S$ is an one point set. – P.. Dec 18 '12 at 10:22

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