# 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.

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