Proving that a surjective homomorphism can help generate a finitely generated k-algebra

Question

I am trying to understand a proof in Dummit and Foote. It can be found in Chapter 15, Section 1, Corollary 5.

The corollary is

The ring $R$ is a finitely generated $k$-algebra iff there is some surjectiv $k$-algebra homomorphism

$\phi: k[x_1, \ldots, x_n] \rightarrow R$

that is the identity map on $k$.

In the proof, the justification for right to left is

"Given a surjective homomorphism $\phi$, the images of $x_1, \ldots , x_k$ under $\phi$ then generate $R$ as a $k$-algebra."

I do not understand how the images of these generate $R$ as a $k$-algebra. I can see it is thus a Noetherian ring, but I can't bridge that to $k$-algebra.

Answer 1

The images of $x_1, \dots, x_n$ generate some $k$-algebra. This is because the map $\phi$ induces a multiplication on its image by pulling back: $$ \phi(x_1)\phi(x_2) := \phi(x_1x_2). $$ But since $\phi$ is surjective, its image is all of $R$, so $\phi$ induces a $k$-algebra structure on $R$.