Does R[x] \cong S[x]R[x] \cong S[x] imply R \cong SR \cong S?

This is a very simple question but I believe it’s nontrivial.

I would like to know if the following is true:

If R and S are rings and R[x] and S[x] are isomorphic as rings, then R and S are isomorphic.

Thanks!

If there isn’t a proof (or disproof) of the general result, I would be interested to know if there are particular cases when this claim is true.

Answer

Here is a counterexample.

Let R=\dfrac{\mathbb{C}[x,y,z]}{\big(xy – (1 – z^2)\big)}, S=\dfrac{\mathbb{C}[x,y,z]}{\big(x^2y – (1 – z^2)\big)}. Then, R is not isomorphic to S but, R[T]\cong S[T].

In many variables, this is called the Zariski problem or cancellation of indeterminates and is largely open. Here is a discussion by Hochster (problem 3).

Attribution
Source : Link , Question Author : Richard G , Answer Author : Batominovski

Leave a Comment