Let p be a prime and let (mod $p$)$ : Z[x] \mapsto Z_p[x]$ be the mod-p map which sends any polynomial...
$f(x) = a_0 + a_1x + a_2x^2 + · · · + a_nx^n \in Z[x]$
... to the polynomial...
$f(x)$(mod $p$) $= a_0 + a_1x + a_2x^2 + · · · + a_nx^n$ where $a_i := a_i$ (mod $p$).
I must show whether or not the map is a ring homomorphism, and whether or not it's onto.
My first instinct is to create another function $g(x)= b_0 + b_1x +b_2x^2 + ... + b_nx^n$ to check if the map is closed under addition and multiplication. Is this a correct approach?