Assume $S$ to be all continuous functions from $[0,1]$ to $\mathbb R$. I know by compactness of $[0,1]$ it follows that all maximal ideals of $S$ have the form $M_{x_0}=\{f\in S \mid f(x_0)=0\}$.Does there exist any non maximal prime ideal of $S$.I tried with some ideals of S but could not find any.please give me some idea how can I construct this?
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Please also add to your list of "things to do before asking a question" a thorough search of our questions that already exist. This is the third question you've asked recently that could be considered a duplicate. I found this one by googling "site: math.stackexchange.com maximal prime ideal continuous functions". – rschwieb Sep 18 '14 at 19:17
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@rschwieb, Good point! I guess I should have searched before answering, as well. In any case, I don't think my example has been written down as an answer to the other question! – Stephen Sep 18 '14 at 19:20
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@rschwieb,My mistake! although i din't searched on stack exchange but i searched on google and i din't find any satisfacory answer. – Arpit Kansal Sep 18 '14 at 19:24
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@Stephen Unless I'm just fooled by similarities, I think your approach was already used at this other duplicate of this question. – rschwieb Sep 18 '14 at 19:25
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@ArpitKansal Actually the site search worked pretty well in this case, too. I put in "prime maximal continuous function" and the duplicate was fourth on the list. – rschwieb Sep 18 '14 at 19:26
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@rschwieb Thanks, but you're giving me too much credit! My idea definitely doesn't work. – Stephen Sep 18 '14 at 19:46