Discussion
Which of the following rings is not Noetherian ?
Answer: A
The ring that is not Noetherian is \(\mathbf{R}[x_{1}, x_{2}, \ldots]\).
In the polynomial ring in infinitely many variables, the chain of ideals
\[ \begin{aligned} (x_{1}) &\subset (x_{1}, x_{2}) \\ &\subset (x_{1}, x_{2}, x_{3}) \subset \cdots \end{aligned} \]
is strictly increasing and never stops, so the ascending chain condition fails. The ideal generated by all the variables is not finitely generated.
\(\mathbf{Z}\) and \(\mathbf{Z}[i]\) are PIDs, hence Noetherian, and \(\mathbf{Z}_{205}[x]\) is Noetherian by Hilbert's basis theorem, since \(\mathbf{Z}_{205}\) is a finite ring and so Noetherian.
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