If the polynomial ring $R[x]$ is Noetherian, then $R$ is noetherian.
abstract-algebraring-theorynoetherian
asked 2012-11-19
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Any factor ring of a noetherian ring is noetherian. Since $x$ generates a two-sided ideal of $R[x]$ and $R[x]/(x)$ is isomorphic to $R$, then $R$ is noetherian. – 2012-11-19