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Let the ring $\varepsilon_n=\lbrace f\colon (\mathbb{R}^{n},0)\longrightarrow \mathbb{R}: f \quad\textit{is map germ}\rbrace$ and maximal ideal $\mathcal{M}_{n}=\lbrace f\in\varepsilon_{n}:[f]_{0}=0\rbrace$, that is, $\varepsilon$ is the module of the smooth functions evaluated at zero and $\mathcal{M}_n$ is the ideal of the functions evaluated at zero are vanished.

I must give a counterexample to show that $\varepsilon_n$ isn't Notherian. I have consulted and a suggestion is to use the Nakayama lemma, however I want to show directly from the definition of a Notherian module, that is, to show that it is infinitely generated. Prove that the generated of the following module is infinitely generated or ascending chain, in another post, they suggested that $$\langle f_n\rangle=\left\langle \frac{1}{x^n}e^{-\frac{1}{x^2}}\right\rangle_{n\in\mathbb{N}}$$

I think it is infinitely generated since, if we consider the case $n=1$ and its respective derivative: \begin{equation} f_1'=-\frac{1}{x}\left(\frac{1}{x}e^{-\frac{1}{x^2}}\right)+\frac{2}{x^3}\left(\frac{1}{x} e^{-\frac{1}{x^2}}\right) \end{equation} and if I calculate the $ n -th $ derivative I have to tend to zero, and so we can see for any $ n $ however I don't know how to justify this formally

logarithm
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  • I think You cannot prove Noetherianity by showing it is infinitely generated. Everything which is finitely generated can be also infinitely generated with other set of generators. Or I did not understand Your intention. – robin3210 May 11 '21 at 16:42
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    Assume that it can be generated by $f_1,f_2,...,f_m$ and look at the smallest $k$ such that all those generators multiplied by $x^k$ is a germ of a polynomial times $e^{-1/x^2}$. For such $k$ the same will be true for all elements of the ideal generated by $f_1,f_2,...,f_m$. Then show that $x^{-(k+1)}e^{-1/x^2}$ doesn't have that property. Therefore it is in your ideal but not in the ideal generated by $f_1,f_2,...,f_m$. – plop May 11 '21 at 16:52
  • @plop I was looking at other questions in this forum, and I found this one https://math.stackexchange.com/questions/1239569/is-the-ring-of-germs-of-c-infty-functions-at-0-noetherian, I will try to try your comment – logarithm May 11 '21 at 20:20

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