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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.

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This was recently asked and answered at https://math.stackexchange.com/questions/13504/does-rx-cong-sx-imply-r-cong-s As answered there, it is possible to find two non-isomorphic commutative rings whose polynomial rings in one variable are isomorphic. An example is given in http://www.ams.org/journals/proc/1972-034-01/S0002-9939-1972-0294325-3/home.html

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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.

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