Definition 1

The determinant of an endomorphism $u$ of a free $A$-module $M$ of finite dimension $n$ (II, § 7, no. 2, Corollary to Proposition 3 and Remark 1), denoted by $\det u$, is the scalar $\lambda$ such that $\bigwedge^n(u)$ is the homothety of ratio $\lambda$.

By formula (4) of § 7, no. 2, $\det u$ is the unique scalar such that

(1)

for every sequence $(x_i)_{1 \leq i \leq n}$ of $n$ elements of $M$. If $\det(u) = 1$, $u$ is said to be unimodular.