MHB Prove $R\ncong R\left[x\right]$ for Noetherian Ring

Thread starter chuyenvien94
Start date Nov 2, 2013
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In a commutative Noetherian ring with identity, it is proven that the ring $R$ is not isomorphic to the polynomial ring $R[x]$. An example illustrates that this result fails if $R$ is not Noetherian, highlighting the importance of the Noetherian condition. Participants discuss various attempts to prove the isomorphism and identify common mistakes in reasoning. The conversation emphasizes the necessity of understanding the properties of Noetherian rings in relation to polynomial rings. Ultimately, the distinction between Noetherian and non-Noetherian rings is crucial for this theorem.

Nov 2, 2013

chuyenvien94

Let $R$ be a commutative Noetherian ring with identity. Prove that $R\ncong R\left[x\right]$ and give an example that the result is not true if $R$ is not Noetherian.

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Nov 2, 2013

MarkFL

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MHB

Can you show what you have tried so our helpers know where you are stuck and/or what mistake(s) you may be making?

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