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 and M an A-module. Suppose given in each of the $A_{f_i}$-modules $M_{f_i}$ an element $z_i$ such that, for $i \neq j$, the canonical images of $z_i$ and $z_j$ in $M_{f_if_j}$ are equal, Show that there exists a unique $z \in M$ such that, for all $i$, the canonical image of $z$ in $M_{f_i}$ is equal to $z_i$. (Note 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 and P a projective A-module of rank 1. Show that the ring $\operatorname{End}_A(P)$ is canonically isomorphic to A (use Theorem 3).

(c) Let A be a ring and P a projective A-module of rank $n$. For every endomorphism $u$ of P, there corresponds canonically to $A^n u$, by (b), an element $\det(u)$ of A, called the determinant of $u$, such that $\det(u \circ v) = (\det u)(\det v)$ for two endomorphisms $u, v$ of P. The element $\chi_u(T) = \det(T \cdot 1 - u)$ of $A[T]$ is called the characteristic polynomial of $u$ (cf. no. 3, Proposition 4); show that $\chi_u$ is a monic polynomial of degree $n$ and that $\chi_u(u) = 0$ (use (a) and § 3, no. 3, Theorem 1). Deduce that, for $u$ to be bijective, it is necessary and sufficient that $\det(u)$ be invertible in A. If A is an integral domain, $\chi_u(T) = \prod_{i=1}^n (T - \alpha_i)$ the decomposition into the linear factors of $\chi_u$ in an algebraically closed extension of the field of fractions of A, and $\chi_{q(u)}(T) = \prod_{i=1}^n (T - q(\alpha_i))$ for every polynomial $q \in A[T]$. Generalize similarly Proposition 14 of Algebra, Chapter VII, § 5, no. 6.