Lemma 1

Let $A$ be a ring, $R$ an algebra (not necessarily commutative) over $A$ and $x$ an element of $R$. Let $M$ be a faithful module over $A[x]$ which is a finitely generated $A$-module. If $q$ is an ideal of $A$ such that $xM \subset qM$, then $x$ is a root of a monic polynomial with coefficients in $A$ all $\notin$ whose coefficients other than the dominant coefficient belong to $q$.

Let $(u_i)_{1 \leq i \leq n}$ be a finite family of elements of $M$ such that $M = \sum_{i=1}^n Au_i$. For all $i$, there exists by hypothesis a finite family $(q_{ij})_{1 \leq j \leq n}$ of elements of $q$ such that

Therefore (Algebra, Chapter III, § 8) if $d$ is the determinant of the matrix $(q_{ij} - \delta_{ij}x)$ with elements in $A[x]$ ($\delta_{ij}$ denoting the Kronecker index), then $du_i = 0$ for all $i$ and hence $dM = 0$; as $M$ is assumed to be a faithful $A[x]$-module, necessarily $d = 0$. This means that $x$ is a root of the polynomial $\det(q_{ij} - \delta_{ij}X)$ in $A[X]$ which, up to a sign, is a monic polynomial whose coefficientsother than the dominant coefficient belong to $q$.