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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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