Proposition 1

Let E be a right A-module. The following three properties are equivalent:

(a) E is flat for A, (in other words, for every left ideal $a$ of A, the canonical homomorphism $E \otimes_A a \to E \otimes_A A, = E$ is injective). (b) E is M-flat for every left A-module M. (c) For every exact sequence of left A-modules and homomorphisms

the sequence

is exact.

It is immediate that (b) implies (a). Conversely, suppose that (a) holds; by Lemma 5 of no. 2, E is flat for every free left A-module; as every left A-module is isomorphic to a quotient of a free module (Algebra, Chapter 11, § 1, no. 11, Proposition 20), it follows from Lemma 4 of no. 2 that E is flat for M.

We show that (c) implies (b). If $v : M' \to M$ is an injective homomorphism, the sequence $0 \to M' \xrightarrow{v} M$ is exact; by (c) the sequence

is exact; this means that $1 \otimes v$ is injective, in other words, E is M-flat.

Finally, the implication (b) $\Rightarrow$ (c) is a consequence of the following more precise lemma: