Lemma 5

Let $M$ and $N$ be two supplementary submodules in $E$ and $p : E \to M, q : E \to N$ the canonical projections. The diagram

is commutative and the vertical arrows are bijective.

They are proved as Lemmas 1 and 2, taking account of § 1, no. 6, Corollary 2 to Proposition 6, § 2, no. 1, Proposition 1 and § 3, no. 7, Proposition 7.

The remainder of the proof then proceeds as in (i) and is reduced to the case where $E = A_s$; the two sides of (7) are then canonically identified with $G \otimes_B F$ and $\nu$ becomes the identity.

In particular take $B = A$ and $G$ the $(A, A)$-bimodule ${}_sA_d$ ($§ 3$, no. 4), so that the right $A$-module $\mathrm{Hom}_A(E, {}_sA_d)$ is just the dual $E^*$ of $E$ and $({}_sA_d) \otimes_A F$ is canonically identified with $F$ ($§ 3$, no. 4, Proposition 4). Homomorphism (7) then becomes a canonical $\mathbf{Z}$-homomorphism

and $\theta(x^* \otimes y)$ is the linear mapping of $E$ into $F$

Remark (1). The characterization of projective $A$-modules given in $§ 2$, no. 6, Proposition 12, can also be expressed as follows: for a left $A$-module $E$ to be projective, it is necessary and sufficient that the canonical homomorphism

be such that $1_E$ belongs to the image of $\theta_E$.