TITLE: Is a finite inverse limit of noetherian rings noetherian?
QUESTION [7 upvotes]: Let $\{A_i\}$ be an inverse system of (commutative, unital) Noetherian rings with a finite index set. Is $\varprojlim A_i$ also a Noetherian ring?

REPLY [6 votes]: The answer is no.
Let $\varphi : k[x,y] \to k[x,y], f \mapsto f(x,0)$ and
$$A = \{ f \in k[x,y] ~|~ f(x,0) \in k \} = \varphi^{-1}(k).$$
$A$ is well known to be non-noetherian - $(y,xy,x^2y,x^3y, \dotsc)$ is not finitely generated - but it fits in the following cartesian square (the horizontal arrows are inclusions):
$$\require{AMScd}
\begin{CD}
A @>>> k[x,y]\\
@VV\varphi V @VV\varphi V \\
k @>>> k[x,y]
\end{CD}$$