Proposition 1

If the exact sequence of linear mappings

(3)

splits (in other words, if $u(E')$ is a direct factor of E) the sequence

(4)

is exact and splits. Conversely, if, for every A-module F, sequence (4) is exact, sequence (3) splits.

If the exact sequence (3) splits, there exists a linear retraction $u' : E \to E'$ associated with $u$ (§ 1, no. 9, Proposition 15); if

the fact that $u' \circ u$ is the identity implies that $\bar{u} \circ \bar{u}'$ is the identity ($§ 1$, no. 2, formula (10)) and hence the first assertion follows from $§ 1$, no. 9, Proposition 15. Conversely, suppose sequence (4) is exact for $F = E'$. Then there exists an element $f \in \mathrm{Hom}(E, E')$ such that $f \circ u = 1_{E'}$, and the conclusion follows from $§ 1$, no. 9, Proposition 15.

Note that the first assertion of Proposition 1 can also be considered as a special case of $§ 1$, no. 6, Corollary 1 to Proposition 6, canonically identifying $\mathrm{Hom}(E', F) \oplus \mathrm{Hom}(E'', F)$ with $\mathrm{Hom}(E' \oplus E'', F)$ by means of the $\mathbf{Z}$-linear mapping $\mathrm{Hom}(p', 1_F) + \mathrm{Hom}(p'', 1_F)$, where $p': E' \oplus E'' \to E'$ and $p'': E' \oplus E'' \to E''$ are the canonical projections.