Is there any non-trivial homomorphism from $R_1=\mathbb{Z}[x]/\langle x^n-1 \rangle$ to $R_2=\mathbb{Z}_m[x]/\langle x^n+1 \rangle$, for all $m>1$ (no necessarily prime)? (Note that the elements in $R_1$ can have negative values, while the ones in $R_2$ only non-negatives).
As there is a finite number of elements in $R_2$ and an infinite (countable) number of elements in $R_1$, I know that there could not be any isomorphism. But is there a way to construct a homomorphism?