Lemma 6

If $M' \xrightarrow{v} M \xrightarrow{w} M''$ is an exact sequence of left A-modules and if E is an $M''$-flat right A-module, the sequence

is exact.

We use the notation $T(Q)$ and $T(v)$ in the same sense as in the proof of Lemma 4 of no. 2. We write $M_1'' = w(M)$ and let $i : M_1'' \to M''$ be the canonical injection and $p$ the mapping of M to $M_1''$ with the same graph as w. The sequence $M' \xrightarrow{v} M \xrightarrow{p} M_1'' \to 0$ is exact and it follows from Lemma 1 of no. 1 that the sequence

is exact. Moreover, as E is $M''$-flat, the mapping $T(i) : T(M_1'') \to T(M'')$ is injective, and as $T(i) \circ T(p) = T(w)$, the sequence

is exact. (§ 1, no. 3.)