Assume you have two affine varieties $X$ and $Y$. A morphism $\phi$ between them induces a morphism between the k-algebras of regular functions (functions on them that are locally the quotients of polynomials) $\phi^*:\mathcal{O}_Y(Y) \rightarrow \mathcal{O}_X(X)$. Assuming $\phi^*$ is surjective, what can one say about its kernel?
It is obviously an ideal of $\mathcal{O}_Y(Y)$ but I was wondering if it has to be radical or even prime. I was wondering maybe there is an analogous of the nullstellensatz for this setting. Thanks to anyone who might have any tips.
You don't need surjectivity of $\phi^*$ for this : $O_Y(Y) / \ker(\phi^*)$ is always a subalgebra of $O_X(X)$, and a subring of a reduced ring is reduced. Thus $\ker(\phi^*)$ is a radical ideal of $O_Y(Y)$.
Finally, if $X$ is also irreducible, then $O_X(X)$ is an integral domain and so is $O_Y(Y) / \ker(\phi^*)$, so that $\ker(\phi^*)$ is prime.