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

MHB
Thread starter chuyenvien94
Start date Nov 2, 2013
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In summary, a Noetherian ring is a commutative ring where every ascending chain of ideals stabilizes. Proving that $R\ncong R\left[x\right]$ for Noetherian rings is important in understanding their structure and has implications in various areas of mathematics, including algebraic geometry and commutative algebra. The proof is typically done by constructing a homomorphism and showing it is not an isomorphism. This result also has consequences in representation theory, algebraic number theory, and computer science.

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?

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

1. What is the definition of a Noetherian ring?

A Noetherian ring is a commutative ring in which every ascending chain of ideals stabilizes, meaning that there is no infinite sequence of ideals I₁ ⊂ I₂ ⊂ I₃ ⊂ ... that continues indefinitely.

2. Why is it important to prove that $R\ncong R\left[x\right]$ for Noetherian rings?

This proof is important because it shows that the ring R does not have the same structure as the polynomial ring R[x]. This result can be used to better understand the properties and structure of Noetherian rings.

3. What is the significance of proving that $R\ncong R\left[x\right]$ specifically for Noetherian rings?

Noetherian rings are important in many areas of mathematics, such as algebraic geometry and commutative algebra. Therefore, proving this result for Noetherian rings has implications for these fields and can help to further our understanding of Noetherian rings.

4. How is the proof of $R\ncong R\left[x\right]$ for Noetherian rings typically done?

The proof typically involves constructing a homomorphism from R to R[x] and showing that it is not an isomorphism. This is usually done by considering the ideals generated by certain elements in R and R[x] and showing that they are not equal.

5. Are there any other important consequences of proving $R\ncong R\left[x\right]$ for Noetherian rings?

Yes, this result has important consequences for other areas of mathematics, such as representation theory and algebraic number theory. It also has applications in computer science, particularly in coding theory and cryptography.