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.



Can we deduce that two rings $R_1$ and $R_2$ are isomorphic if their polynomial ring are isomorphic?
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>


