Is $\mathbb{Z}[x]/(x^2-2)$ isomorphic to $\mathbb{Z}[x]/(x^2-3)$?
It seems that an isomorphism can hardly be defined. But I cannot find any property that $\mathbb{Z}[x]/(x^2-2)$ has but $\mathbb{Z}[x]/(x^2-3)$ not or vice versa, so I cannot say those to quotient ring can not be isomorphic.