Let $\Bbb{Z}[X,Y]$ be the polynomial ring. I know that every evaluation ring hom $f: \Bbb{Z}[X,Y] \to \Bbb{Z}$ is determined by where you send $X$ and $Y$. Let $I = (X^2 - Y^3)$ for example, but it could be generated by any collection of polynomials.
Let $\pi : \Bbb{Z}[X, Y] \to \Bbb{Z}/I$ be the canonical map. I know that if $\ker f \supset \ker \pi$, then we can make a well-defined map $g : \Bbb{Z}[X,Y]/I \to \Bbb{Z}$ simply by setting $g(\pi(x)) := f(x)$. So in a sense there is an evaluation map from $\Bbb{Z}[X,Y]/I$ onto $\Bbb{Z}$.
My question is the converse, given a surjective "evaluation map" $g : \Bbb{Z}[X,Y]/I \to \Bbb{Z}$ does there exist some evalution map $f:\Bbb{Z}[X,Y] \to \Bbb{Z}$ such that $f = g \circ \pi$?
Firstly, I'm not sure if evaluation map from the quotient is even a well-formed concept.