Proposition 2

If the exact sequence

(7)

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

(8)

is exact and splits. Conversely, if sequence (8) is exact for every A-module E, the exact sequence (7) splits.

The first assertion follows from the fact that

is canonically identified with $\operatorname{Hom}(E, F' \oplus F'')$ by means of the $\mathbf{Z}$-linear mapping $\operatorname{Hom}(1_E, j') + \operatorname{Hom}(1_E, j''), j' : F' \to F' \oplus F''$ and $j'' : F'' \to F' \oplus F''$ being the canonical injections (§ 1, no. 6, Corollary 1 to Proposition 6). Conversely, if sequence (8) is exact for $E = F''$, there is an element $g \in \operatorname{Hom}(F'', F)$ such that $v \circ g = 1_{F''}$ and the conclusion follows from § 1, no. 9, Proposition 15.

Remark (2). The results of this no. are valid without modification for all commutative groups with operators.