Exercise 9, § 1

Show that in an A-module E, if $(F_n)_{n \geq 1}$ is a strictly increasing sequence of finitely generated submodules, the submodule F of E the union of the $F_n$ is not finitely generated. Deduce that if E is an A-module which is not finitely generated, there exists a submodule F of E such that $\gamma(F) = \aleph_0$ ($= \mathrm{Card}(\mathbf{N})$).