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Let $a,b,c,d$ be real numbers with $a

(A) $S$ is NOT an of $C[a,b]$.

(B) $S$ is an ideal of $C[a,b]$ but NOT a prime ideal of $C[a,b]$.

(C) $S$ is a prime ideal of $C[a,b]$ but NOT a maximal ideal of $C[a,b]$.

(D) $S$ is a maximal ideal of $C[a,b]$.

I am submitting my answer below and looking for better solutions. Thanks.

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    I think, given that you've answered your own question (i.e., adequately provided context in any reasonable sense of the word), it should absolutely be open. But maybe (in truth, I have no idea) this sort of thing would be less likely to happen if you put your proposed solution in the body of the question. – 2017-01-07
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    I had faced a similar problem earlier. It was discussed [here](http://meta.math.stackexchange.com/questions/11214/qa-style-construed-as-help). It's not a good use of time when we have to defend ourselves again and again. – 2017-01-07

1 Answers 1

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Differnce of two zero functions is a zero function. And zero function, after product with any other function, would remain a zero function. So, S is an ideal.

If we take two functions from $C[a,b], f$ and $g$. And $fg(x)=0$ for all $x\in[c,d]$. If either of $f$ or $g$ is non-zero then the other would belong to $S$. If both are zero at different intervals say, $f(x)=0$ for all $x\in[c,e]$. $e$ is any point in $[c,d]$. And $g(x)=0$ for all $x\in[f,d]$. $f$ is any point in $[c,e]$. So, $S$ is not a prime ideal.

Therefore, the correct option is B.