Exercise 6, § 1

Given an A-module E, let $\gamma(E)$ denote the least of the cardinals of the generating systems of E.

(a) If $0 \to E \to F \to G \to 0$ is an exact sequence of A-modules, then

Give an example of a product $F = E \times G$ such that $\gamma(E) = \gamma(F) = \gamma(G) = 1$ (take E and G to be quotients of $\mathbf{Z}$).

(b) If E is the sum of a family $(F_\lambda)_{\lambda \in L}$ of submodules, then $\gamma(E) \leq \sum_{\lambda \in L} \gamma(F_\lambda)$.

(c) If M and N are submodules of a module E, show that