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. 

I found the paper Isomorphic polynomial rings by Brewer and Rutter that discusses
related matters. They cite a forthcoming paper by Hochster
which proves there are non-isomorphic
commutative integral domains $R$ and $S$ with $R[x]\cong S[x]$.
Added
Hochster's paper is
M. Hochster, Nonuniqueness of coefficient rings in a polynomial ring,
Proc. Amer. Math. Soc. 34 (1972), 81-82, and is freely available.
