TITLE: Can we deduce that two rings $R_1$ and $R_2$ are isomorphic if their polynomial ring are isomorphic?
QUESTION [7 upvotes]: Given two rings $R_1$ and $R_2$ (with or without identity: it's not specified). If $R_1[x]$ is isomorphic to $R_2[y]$ (No such requirement that the isomorphism sends the constant terms to constant terms), can we deduce that $R_1 \cong R_2$?
I feel there might be a counterexample but it's quite hard to find one.

REPLY [4 votes]: Let X be an affine variety with two non-isomorphic vector bundles V and W that become isomorphic after adding a trivial line bundle to each.  Then the coordinate rings of the total spaces of V and W should yield a counterexample.   (Though you might have to do some extra work to verify that the total spaces of V and W are non-isomorphic as varieties.)  
The counterexample cited by Tobias takes X to be the 2-sphere, V the tangent bundle to X, and W the trivial plane bundle over X.