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Let $A=C(0,1)$ be the ring of continuous real valued functions on the open interval $(0,1)$. It is not too difficult to show that if $\mathfrak{m}\subseteq A$ is a maximal ideal with residue field $A/\mathfrak{m}\simeq \mathbb{R}$ then $\mathfrak{m}=\mathfrak{m}_a:=ker(ev_a)$ where for $a\in(0,1)$, $ev_a:A\rightarrow\mathbb{R}$ is the evaluation map at $a$. It is easy to show that there are maximal ideals not of the for $\mathfrak{m}_a$. For instance one may look at the ideal

$$ I=\{f\in A:\exists n\in\mathbf{Z}_{\geq 1}, f\equiv 0\;\;\mbox{on $(0,1/n]$}\} $$ Then $I$ is not contained in any $\mathfrak{m}_a$ but by Zorn's lemma it is contained in some maximal ideal $\mathfrak{M}$. So here are two natural questions on the ring $A$ for which I don't have an answer:

Q1: Do we have a structure theorem for the possible residue fields of maximal ideals of $A$.

Q2: How does one show the existence (or construct) of a prime ideal of $A$ which is not maximal?

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  • $\begingroup$ One might try to construct a multiplicatively closed set with certain bad properties. I don't think the set of functions with finitely many zeroes works, but it might be a start. I think one might have to classify the maximal ideals first. This might be impossible. $\endgroup$ – Will Sawin Dec 9 '11 at 15:44
  • $\begingroup$ @Will: sharing the fact that your first thoughts have led nowhere seems not very helpful to me. $\endgroup$ – user2035 Dec 9 '11 at 16:39
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    $\begingroup$ Q1 is the same question as mathoverflow.net/questions/3871/…, where the answerer noted that the residue fields of maximal ideals not arising from points are necessarily more exotic than $\mathbf{C}$ (or $\mathbf{R}$ in the context of the current post). $\endgroup$ – Jared Weinstein Dec 9 '11 at 18:34
  • $\begingroup$ Hi Hugo - you might want to have a look at work of Hung Le Pham, or at least at some of the prior work that he builds on, see homepages.ecs.vuw.ac.nz/~hlpham $\endgroup$ – Yemon Choi Dec 9 '11 at 20:47
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Take any free ultrafilter $U$ on $(0,1)$, and let $m_U$ be the set of functions which are $0$ "almost everywhere", i.e., on a set in the ultrafilter. It seems to me that is a prime ideal, which is in general not maximal. If all ultrafilter sets have, say, the number 1/2 as a limit point, then $m_U$ is properly contained in the ideal of functions vanishing at 1/2.

Why is it an ideal? That looks easy. E.g., if you have two functions vanishing on $A_1$, $A_2$ respectively, then their sum vanishes on $A_1\cap A_2$, which is again in the ultrafilter.

Why is it prime? Let $f_3:=f_1\cdot f_2$. Let $A_i:= \{x:f_i(x)=0\}$. Then $f_3\in m_U$ iff $A_3 \in U$ iff $A_1\cup A_2\in U$ iff $A_1\in U $ or $A_2\in U$ iff one of $f_1$, $f_2$ is in $m_U$.

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  • $\begingroup$ Could you please indicate how to construct an ultrafilter belonging to a non-maximal prime ideal (or show its existence)? $\endgroup$ – user2035 Dec 9 '11 at 17:28
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    $\begingroup$ @a-fortiori: Extend the filter generated by the sets $((1-1/n)/2,0)\cup(0,(1+1/n)/2)$ to an ultrafilter $U$. Note that that if $f \in m_U$ then $f(1/2) = 0$ by continuity, but $m_U$ is not the ideal of all functions that vanish at $1/2$ since $f(x) = x-1/2$ is not in $m_U$. $\endgroup$ – François G. Dorais Dec 9 '11 at 18:45
  • $\begingroup$ (Of course, I meant the sets $((1-1/n)/2,1/2)\cup(1/2,(1+1/n)/2)$.) $\endgroup$ – François G. Dorais Dec 9 '11 at 18:50
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For the existence part :

Hint : Let S denote the collection of all polynomials in $C(0,1)$ . Consider $J:=C(0,1) \text{ \ } S$ and all the ideals contained in it. Observe that under usual set inclusion , $(J,\subseteq)$ is a poset. Consider an increasing chain and argue by Zorn's lemma.

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