Definition 41.9.1. Flatness of modules and rings.

A module $N$ over a ring $A$ is said to be

*flat*if the functor $M \mapsto M \otimes _ A N$ is exact.If this functor is also faithful, we say that $N$ is

*faithfully flat*over $A$.A morphism of rings $f : A \to B$ is said to be

*flat (resp. faithfully flat)*if the functor $M \mapsto M \otimes _ A B$ is exact (resp. faithful and exact).

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