Discussion
Which of the following rings is not noetherian ?
Answer: A
The non-Noetherian ring is \(\mathbf{Q}[x_{1}, x_{2}, x_{3}, \ldots]\).
With infinitely many variables, the chain \((x_{1}) \subset (x_{1}, x_{2}) \subset (x_{1}, x_{2}, x_{3}) \subset \cdots\) increases strictly and never stops, so the ascending chain condition fails.
The others are Noetherian. \(\mathbf{Z}\) is Noetherian, so \(\mathbf{Z}[x]\) is, by Hilbert's basis theorem. \(\mathbf{Z}[\sqrt{2}]\) is a finitely generated \(\mathbf{Z}\)-module (indeed a Euclidean domain). And \(\mathbf{R}[x, y]\) is Noetherian, so every quotient of it, including \(\mathbf{R}[x, y]/(x^{2} + y^{2} + 1)\), is Noetherian.
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