TITLE: Does $R[x] \cong S[x]$ imply $R \cong S$?
QUESTION [141 upvotes]: 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.

REPLY [23 votes]: 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