Proposition 17

*Let $A$ be a ring, $B$ a (commutative) $A$-algebra and $X$ a square matrix of order $n$ over $B$; the following properties are equivalent:

(a) $X$ is integral over $A$.

(b) There exists a finitely generated sub-$A$-module $M$ of $B^n$ such that $X.x \in M$ for all $x \in M$ and $M$ is a system of generators of the $B$-module $B^n$.

(c) The coefficients of the characteristic polynomial of $X$ are integral over $A$.*

If $\chi(T) = \det(T.1 - X)$ is the characteristic polynomial of $X$, the Cayley-Hamilton Theorem shows that $\chi(X) = 0$ (Algebra, Chapter VII, § 5, no. 4, Remark 1) and, as $\chi$ is a monic polynomial, (c) implies (a) by no. 1, Proposition 6.

Suppose in the second place that (a) holds. If $(e_i)_{1 \leq i \leq n}$ is the canonical basis of $B^n$, the sub-$A$-module $M$ of $B$ generated by the $X^k.e_i$ ($1 \leq i \leq n, k \geq 0$) is a finitely generated $A$-module, since the $A$-algebra $A[X]$ is a finitely generated $A$-module (no. 1, Theorem 1); as $M$ contains the $e_i$, it is seen that (a) implies (b); the converse is a consequence of no. 1, Theorem 1, condition (E_{III}).

Finally let us prove that (a) implies (c); as $X$ is integral over $A$ and a fortiori over the polynomial ring $A[T]$, $T.1 - X$ is also integral over $A[T]$ and by no. 3, Proposition 12 the problem is seen to reduce (by replacing $X$ by $T.1 - X$ and $A$ by $A[T]$) to proving that, if $X$ is integral over $A$, $d = \det(X)$ is an element of $B$ which is integral over $A$. Now, we have seen above that the endomorphism $u$ of $B^n$ defined by $X$ leaves stable a finitely generated sub-$A$-module $M$ containing the $e_i$; the $n$-vectors $x_1 \wedge x_2 \wedge \ldots \wedge x_n$, where $x_i \in M$ for $1 \leq i \leq n$, therefore generate in $\bigwedge^n(B^n)$ a finitely generated sub-$A$-module containing $e_1 \wedge e_2 \wedge \ldots \wedge e$, and which is stable under $\bigwedge^n u$, in other words under the homothety of ratio $d$; as the annihilator of

in B reduces to 0, condition (E_{III}) of no. 1, Theorem 1 proves that d is integral over A.