Theorem 2

Let $A$ be a ring, $F', F, F''$ three $A$-modules and $u: F' \to F, v: F \to F''$ two homomorphisms. For the sequence

to be exact, it is necessary and sufficient that, for every $A$-module $E$, the sequence

(where $\bar{u} = \mathrm{Hom}(1_E, u), \bar{v} = \mathrm{Hom}(1_E, v)$) be exact.

Suppose that the sequence (5) is exact. Note first that

(II, $§ 1$, no. 2, formula (10)) since $v \circ u = 0$. The image of $\mathrm{Hom}(E, F')$ under $\bar{u}$ is therefore contained in the kernel $N$ of $\bar{v}$; let $f$ be the homomorphism of the $\mathbf{Z}$-module $\mathrm{Hom}(E, F')$ into $N$ whose graph is equal to that of $\bar{u}$; it is necessary to prove that $f$ is bijective and hence to define a mapping $g: N \to \mathrm{Hom}(E, F')$ such that $f \circ g$ and $g \circ f$ are the identity mappings. For this, let $w$ be an element of $N$, that is a linear mapping $w: E \to F$ such that $v \circ w = 0$. The latter relation is equivalent to $w(E) \subset \mathrm{Ker}(v) = u(F')$ by hypothesis, hence, since $u$ is injective, there exists one and only one linear mapping $w': E \to F'$ such that $w = u \circ w'$ and we take $g(w) = w'$; it is immediately verified that $g$ satisfies the desired conditions.

Conversely, suppose that sequence (6) is exact for every $A$-module $E$. As $\mathrm{Hom}(1_E, v \circ u) = \bar{v} \circ \bar{u} = 0$, then $v \circ u \circ w = 0$ for every homomorphism $w: E \to F'$. Taking $E = F'$ and $w = 1_{F'}$, it is seen first that $v \circ u = 0$ and hence $u(F') \subset \mathrm{Ker}(v)$. Now we take $E = \mathrm{Ker}(v)$ and let $\phi: E \to F$ be the canonical injection. Then $\bar{v}(\phi) = v \circ \phi = 0$ by definition and hence there exists $\psi \in \mathrm{Hom}(E, F')$ such that $\phi = \bar{u}(\psi) = u \circ \psi$, which obviously implies $\mathrm{Ker}(v) \subset u(F')$ and completes the proof of the exactness of (5) at $F$. Finally, if $\theta$ is the identity mapping of $\mathrm{Ker}\, u$, then $\bar{u}(\theta) = 0$, hence $\theta = 0$ and $\mathrm{Ker}\, u = \{0\}$, which proves the exactness of (5) at $F'$.