This is similar to this but different. The original argument requires Nakayama's lemma, I wonder if the following still holds:
Let $\varphi:A \rightarrow A$ be a surjective $k$-algebra homomorphism between finitely generated $k$-algebra $A$. Then $\varphi$ is injective.
Is this true? This would be useful in study of coordinate rings.
For commutative rings, this is certainly true. The crucial hypothesis you need is that $A$ is Noetherian, which is true for finitely generated algebras over a field. You have a chain of ideals, $\ker \phi\subset\ker \phi^2\subset\ker \phi^3\subset\cdots$ and thus there is an $n$ such that $\ker \phi^n=\ker\phi^{n+1}=\cdots$. This can not happen unless $\ker\phi=0$ since $\phi$ is surjective.