Corollary 1

Suppose that $H_0$ is a field and that $H$ is a finitely generated $H_0$-algebra. Put $n = \dim(H)$. There exist homogeneous elements $a_1, ..., a_n$ of $H$, all of the same degree $> 0$, such that the $H_0$-homomorphism $\varphi : H_0[X_1, ..., X_n] \to H$ defined by $\varphi(X_i) = a_i, i = 1, ..., n$, is injective and makes $H$ into a finite $H_0[X_1, ..., X_n]$-algebra. If $H$ is generated by $H_1$ as an $H_0$-algebra and if $H_0$ is infinite, the $a_i$ may be supposed to be of degree 1.

By Prop. 6 there exists an $H_0$-homomorphism $\varphi$ of the indicated form which makes $H$ into a finite $H_0[X_1, ..., X_n]$-algebra. By Theorem 1 of § 2, No. 3, one then has

since one has and since $H_0[X_1, ..., X_n]$ is an integral domain, this implies $\mathrm{Ker}\ \varphi = \{0\}$.