Theorem 1

Let $\mathbf{A}$ be a (commutative) ring, $\mathbf{R}$ an algebra over $\mathbf{A}$ (not necessarily commutative) and $x$ an element of $\mathbf{R}$. The following properties are equivalent:

(E₁) $x$ is a root of a monic polynomial in the polynomial ring $\mathbf{A}[X]$. (E₂) The subalgebra $\mathbf{A}[x]$ of $\mathbf{R}$ is a finitely generated $\mathbf{A}$-module. (E₃) There exists a faithful module over the ring $\mathbf{A}[x]$ which is a finitely generated $\mathbf{A}$-module.

Let us show first that (E₁) implies (E₂). Let

be a monic polynomial in $\mathbf{A}[X]$ with $x$ as a root; for every integer $q \geq 0$ let $M_q$ be the sub-$\mathbf{A}$-module of $\mathbf{R}$ generated by $1, x, \ldots, x^n + q$. Then for all $q \geq 1$, whence, by induction on $q$,

(*) The results of this chapter and the next one depend on no Book other than Books I to VI, nor on Chapter I, § 4, nor on Chapter 111, § 5.

We conclude that $A[x]$ is equal to $M_0$ and is therefore a finitely generated $A$-module.

As the commutative ring $A[x]$ is a faithful module over itself, ($E_{II}$) implies ($E_{III}$).

Finally, the fact that ($E_{III}$) implies ($E_1$) will follow from the following more precise lemma: