# injective map of coordinate rings gives dominant map on affine varieties [duplicate]

This is one of the exercises I am struggling on: Suppose $$V$$ and $$W$$ are affine varieties,

Show that a one-to-one map of $$k$$-algebras $$k[W]\rightarrow k[V]$$ corresponds to a dominant map $$V\rightarrow W$$

Here is what I know so far: any polynomial mapping $$V\rightarrow W$$ gives the pullback mapping k-algebra homomorphism $$k[W]\rightarrow k[V]$$, and every k-algebra homomorphism $$k[W]\rightarrow k[V]$$ comes from the pullback of a unique polynomial mapping $$V\rightarrow W$$, but this problem concern not just homomorphism, but more general mapping.

In order to show that there is a dominant map, I think first is to establish at least a map, and I made one as follows: identify both $$k[V]$$ and $$k[W]$$ by $$k[x_1,\cdots ,x_n]/I(V)$$ and $$k[y_1,\cdots ,y_m]/I(W)$$ respectively and let $$\Phi$$ be the injective map. Then the easiest map to construct would be $$\phi =(\Phi ([y_1]),\cdots ,\Phi([y_m]))$$ But the problem about this map is that we can make the domain to be $$V$$ but the codomain is not necessary $$W$$ since $$\Phi$$ only gives a function from $$V$$ to $$k$$

The exercise also come with a seemingly useless hint:

$$\phi :V\rightarrow W$$ gives $$V\rightarrow \overline{\phi (V)}\subseteq W$$

Here is a similar post I found, but it concerns on pullback only, and I don't see a way to generalize to all injective mappings
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