I have seen a few questions similar to this on here but nothing the same as this I don't think, so I'm wondering if it's possible:
Given a surjective polynomial map of algebraic sets $\phi: X \to Y$, is the corresponding $k$-algebra homomorphism $\phi^*: k[X]\to k[Y]$ injective?
Like I said, I have seen a few similar questions asking about surjectivity of $\phi^*$ implying injectivity of $\phi$ but I haven't seen this way round. Any insight would be appreciated!