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Browsing through my notes on Artin rings, I have realized that I don't know an example for this and I wasn't able to google anything relevant.

What is an example for a commutative non-Noetherian local ring $(R,\mathfrak{m})$ with nilpotent maximal ideal, i.e. $\mathfrak{m}^n = 0$ for some $n \in \mathbb{N}$?

Note that in this setting $R$ Noetherian $\Leftrightarrow$ $R$ Artinian.

In this line of thoughts:

Is there a standard name for local rings with a nilpotent maximal ideal?

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    $\begingroup$ Take $R = k[x_1,\ldots]$ with $\mathfrak m = (x_1,\ldots)$ and look at $R/\mathfrak m^2$. (However, this does not work for any maximal ideal that is not finitely generated, because it is possible that $\mathfrak m = \mathfrak m^2$, in which case $R/\mathfrak m^2 = R/\mathfrak m$ is a field. An example of such an $\mathfrak m$ is the maximal ideal in a valuation ring with divisible value group.) $\endgroup$
    – R. van Dobben de Bruyn
    Commented Jun 21, 2020 at 19:43
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    $\begingroup$ In fact in the case $\mathfrak m^2 = 0$ where $R$ contains $k = R/\mathfrak m$, we must have $$R = k \oplus M = \operatorname{Sym}^*(M)/\operatorname{Sym}^{\geq 2}(M)$$ for some $k$-module $M$. This is an example of the type you're looking for if and only if $M$ is not finitely generated. $\endgroup$
    – R. van Dobben de Bruyn
    Commented Jun 21, 2020 at 19:49
  • $\begingroup$ @R.vanDobbendeBruyn: thanks for making it obvious what to look at! If you post it as an answer, I will accept it. $\endgroup$
    – M.G.
    Commented Jun 23, 2020 at 18:44

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