I will state it again. 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 iso. to R_2?
I feel there might be a counterexample but it's quite hard to find one.