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Let $A$ be a commutative ring with unit. How to do the following questions related to Hilbert Basis Theorem? I am quite confused about the proof of Hilbert Basis Theorem.

  1. If $A[x]$ is Noetherian then $A$ is Noetherian.

  2. If $A[x]$ is Noetherian then $A[[x]]$ (power series) is Noetherian.

Shiquan
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  • (1) is the converse of Hilbert's basis theorem. It is very easy to prove. You can then replace (2) with "If $A$ is noetherian then $A [[x]]$ is noetherian", which is proved the same way as Hilbert's basis theorem. – Zhen Lin Oct 15 '13 at 07:10
  • Related for (1): https://math.stackexchange.com/questions/240555/converse-to-hilbert-basis-theorem – Watson Jan 02 '17 at 17:29
  • Related for (2):https://math.stackexchange.com/questions/287113/ – Watson Jan 02 '17 at 17:38

1 Answers1

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  1. Every quotient ring of a noetherian ring is noetherian.
  2. Note that $A[[x]]$ is the completion of the ring $A[x]$ w.r.t. to the ideal $(x)$. The completion of a noetherian ring w.r.t. to some ideal is noetherian (see the book by Atiyah-Macdonald, the chapter on completions). Alternatively, by 1. you know that $A$ is noetherian, and then the usual proof of the Hilbert basis theorem, but looking at "lowest terms" instead of "highest terms" shows that $A[[x]]$ is noetherian.
Martin Brandenburg
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