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.