Proposition 3

In order that the a-module E be flat, it is necessary and sufficient that, for every injective A-homomorphism $u : M' \to M$ of right A-modules, the mapping $u \otimes 1 : M' \otimes_A E \to M \otimes_A E$ be injective.

If E is flat and $u : M' \to M$ is injective, the sequence $0 \to M' \xrightarrow{u} M$ is exact, hence also the sequence $0 \longrightarrow M' \otimes_A E \xrightarrow{u \otimes 1} M \otimes_A E$, and $u \otimes 1$ is injective. Conversely, consider the exact sequence (11); put $M''_1 = v(M)$, and let $i : M''_1 \to M''$ be the canonical injection and $p : M \to M''_1$ the mapping $m \mapsto v(m)$. The sequence $M' \xrightarrow{u} M \xrightarrow{p} M''_1 \longrightarrow 0$ is exact; according to II, p. 58, prop. 5, the sequence $M' \otimes_A E \xrightarrow{u \otimes 1} M \otimes_A E \xrightarrow{p \otimes 1} M''_1 \otimes_A E$ is therefore exact. Moreover, one has $v = i \circ p$, hence $(v \otimes 1) = (i \otimes 1) \circ (p \otimes 1)$; if E satisfies the condition of the statement, then $i \otimes 1$ is injective, hence

and the sequence (12) is exact.