Corollary 6

Let P be a projective left A-module and E, F two right A-modules. For every injective homomorphism $u : E \to F$, the homomorphism

is injective.

There exists a left A-module Q such that $L = P \oplus Q$ is free (§ 2, no. 2, Proposition 4) and $u \otimes 1_L$ is identified (Proposition 7) with

it therefore suffices to prove the corollary when P is free (§ 1, no. 6, Corollary 1 to Proposition 7). The same argument reduces the problem to the case where $P = A_s$, which follows immediately from no. 4, Proposition 4.