Exercise 9, § 5

a) Let A be a ring, $(f_i)_{i \in I}$ a finite family of elements of A generating the ideal A, M an A-module. Suppose that in each of the $A_{f_i}$-modules $M_{f_i}$ an element $z_i$ is given, such that, for $i \neq j$, the canonical images of $z_i$ and $z_j$ in $M_{f_i f_j}$ are equal. Now prove that there exists one and only one $z \in M$ such that, for every $i$, the canonical image of z in $M_{f_i}$ is equal to $z_i$. (Remark that, for every integer $k > 0$, there exists a family $(g_i)_{i \in I}$ of elements of A such that $\sum_i g_i f_i^k = 1$).

b) Let A be a ring, P a projective a-module of rank 1. Now prove that the ring $\mathrm{End}_A(P)$ is canonically isomorphic to A (use th. 3).

c) Let A be a ring, P a projective a-module of rank n. For every endomorphism u of P, there corresponds canonically to $\wedge^n u$, by virtue of b), an element $\det(u) \in A$, called the determinant of u, such that $\det(u \circ v) = (\det u)(\det v)$ for two endomorphisms u, v of A. The characteristic polynomial of u is called the element $\chi_u(T) = \det(T.1 - u)$ of $A[T]$ (cf. n° 3, prop. 4); now prove that $\chi_u$ is a monic polynomial of degree n and that one has $\chi_u(u) = 0$ (use a), and § 3, n° 3, th. 1). Deduce that, in order that u be bijective, it is necessary and sufficient that $\det(u)$ be invertible in A. If A is an integral ring, $\chi_u(T) = \prod_{i=1}^n (T - \alpha_i)$ the decomposition into linear factors of $\chi_u$ in an algebraically closed extension of the field of fractions of A, one has $\chi_{q(u)}(T) = \prod_{i=1}^n (T - q(\alpha_i))$ for every polynomial $q \in A[T]$. Generalize analogously prop. 14 of Alg., chap. VII, § 5, n° 6.