Proposition 21

Every exact sequence of A-modules

in which F is a free A-module, splits (no. 9). To be precise, if $(b_\lambda)_{\lambda \in L}$ is a basis of F and, for each $\lambda \in L$, $a_\lambda$ is an element of E such that $f(a_\lambda) = b_\lambda$, the family $(a_\lambda)_{\lambda \in L}$ is free and generates a supplementary submodule to g(G).

There exists one and only one linear mapping $h : F \to E$ such that $h(b_\lambda) = a_\lambda$ for all $\lambda \in L$ (Corollary 3 to Proposition 17). As h is a linear section associated with f, the proposition follows from I, § 4, no. 9, Proposition 15.