Exercise 10, § 3

Let A be a Zariski ring, m a defining ideal of the topology on A and E a finitely generated A-module. Show that, if $\hat{E}$ is an $\hat{A}$-module admitting a system of generators with r elements, then the A-module E admits a system of generators with r elements (note that $E/mE$ and $\hat{E}/m\hat{E}$ are isomorphic and use Chapter 11, § 3, Corollary 2 to Proposition 4). Is the result valid if E is not assumed to be finitely generated (cf. Exercise 9)? Is it valid if we only assume that A is Noetherian (take $A = \mathbf{Z}$)?