Proposition 11

Let A be a commutative ring, B a ring, $\varphi$ a homomorphism of A to the centre of B, and E and F two A-modules. Suppose that B is a flat A-module and E is finitely generated (resp. finitely presented). Then the canonical homomorphism (10) is injective (resp. bijective).

As $\omega$ is composed of the canonical isomorphism

and the canonical homomorphism (8)

(loc. cit.), the proposition is a consequence of Proposition 10 of no. 9.

Suppose now that A and B are commutative and consider three A-modules $E_1, E_2, E$, and an A-bilinear mapping $f : E_1 \times E \to E_3$. Then there exists one and only one B-bilinear mapping $f_B : E_{1(B)} \times E_{2(B)} \to E_{3(B)}$ such that

for all $x_1 \in E_1, x_2 \in E$, (Algebra, Chapter IX, § 1, no. 4, Proposition 1).

In the statement which follows we shall suppose that B is a flat A-module and, for every submodule E' of $E_i$ ($i = 1, 2, 3$), we shall canonically identify $E'_{(B)}$ with its image in $E_{i(B)}$ (no. 3, Remark 2).