Let $R=\mathbb{Z}[x]$. Calculate $\operatorname{Hom}_R(R/(2x),R/(4))$ as an $R$-module.
My guess is that it's somehow isomorphic to $R/(2)$, as $2$ seems to be the greatest common divisor of $2x$ and $4$, however this is just a guess.
In general, is there a way to calculate $\operatorname{Hom}_{S[x]}(S[x]/I,S[x]/J)$, for some ring $S$ and ideals $I,J\subset S[x]$?