Exercise 10, § 8

We suppose that A is integral; let K be its field of fractions. An ideal a of A is said to be invertible if there exist elements $x_1, ..., x_n$ of K such that $x_i a \subset A$ for $1 \leq i \leq n$ and $\sum_{i} x_i a = A$. a) Now prove that every invertible ideal is of finite type. b) Now prove that a nonzero ideal of A is an a-module projective if and only if it is invertible. c) Let a be an invertible ideal of A, D an a-module divisible. Exercise prove that $\mathrm{Ext}_A^i (A/a, D) = 0$ for $i \geq 1$.