Definition 1

Tag 0DOZ, permanent. Cite it as [Bourbaki, Tag 0DOZ].

One says that the a-module E is flat if, for every exact sequence of right a-modules and homomorphisms

(11) $M' \xrightarrow{u} M \xrightarrow{v} M''$ ,

the sequence of $\mathbf{Z}$ -linear mappings

(12) $M' \otimes_A E \xrightarrow{u \otimes 1} M \otimes_A E \xrightarrow{v \otimes 1} M'' \otimes_A E$

is exact.

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

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