Does $R[x] \cong S[x]$ imply $R \cong S$? 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. 

There's been much work on this problem since the mentioned seminal work in the early seventies. Searching on the buzzword "stably equivalent" should help locate most of it. Below is a helpful introduction from Jon L Johnson: Cancellation and Prime Spectra


