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I'm considering the property of the ring $R:=C^\infty(\mathbb R)/I$, where $I$ is the ideal of all smooth functions that vanish at a neighborhood of $0$. I find that $R$ is a local ring of which the maximal ideal is exactly $(x)$. I also want to determine if $R$ is a Noetherian ring, but I have no idea about how to tackle with it. Can anyone help me? Thanks in advance.

user26857
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Censi LI
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  • http://mathoverflow.net/a/65453 – user26857 Apr 17 '15 at 20:43
  • @user26857 Well, that does show that $R$ cannot be Noetherian since krull intersection theorem fails for $R$. But I still expect a more direct proof, for example, by constructing a infinitely generated ideal of $R$ explicitly... – Censi LI Apr 17 '15 at 20:57

2 Answers2

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I think $\langle e^{-\frac 1{x^{2n}}}\rangle_{n\in\mathbb N}$ is an infinitely generated ideal of $R$.

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    I think that ideal is generated by $e^{-\frac{1}{x^{2}}}$. You want a sequence of functions which tend to zero less and less rapidly, not more and more rapidly. Instead $\langle \frac{1}{x^{n}}e^{-\frac{1}{x^{2}}}\rangle_{n \in \mathbb{N}}$ should work. – Mark Perlman Feb 18 '17 at 22:44
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I think this should work:

Consider the function $f_n$ defined by $$f_n(x) = e^{-1/x^2}\sin(2^n/x).$$ Intuitively, the exponential vanishes so incredibly quickly that the craziness of the sine term causes no problems. More formally, the $n$th derivative (away from zero) of $f_1$ is, by induction, of the form $$e^{1/x^2}[P(1/x)\sin(1/x)+Q(1/x)\cos(1/x)]$$ which is easily seen to be continuous everywhere by the fact that $e^{1/x^2}/x^n$ goes to zero as $x$ goes to zero, for any given $n$, and the trig terms are bound between $-1$ and $1$. $f_n$ should be similar. But the vanishing locus of $f_{n}$ strictly contains that of $f_{n+1}$ (and intersects $(-\varepsilon, \varepsilon)$ infinitely many times), so $$(0) \leq (f_1) \leq (f_1, f_2) \leq \cdots$$ is an infinite ascending chain.

Harrison Smith
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  • What a shame I have a question, I do not understand very well what the ideal generated by $\langle f_1\rangle$ – logarithm May 11 '21 at 15:26