nLab
Lazard's criterion

Statement

Let $R$ be a commutative ring.

Proposition

(Lazard’s criterion)

An $R$-module is a flat module precisely if it is a filtered colimit of free modules.

This is due to (Lazard (1964)). See at flat module for more.

References

• Daniel Lazard, Sur les modules plats C. R. Acad. Sci. Paris 258, 6313–6316 (1964)