Faithfully flat module

Summary

A faithfully flat module is a flat module $M$ such that tensoring with $M$ kills no nonzero module. Then a sequence of modules is exact if and only if it stays exact after tensoring with $M$. A ring map $R \to S$ is faithfully flat when $S$ is faithfully flat as an $R$-module; such maps are the setting of faithfully flat descent. Examples are nonzero free modules, polynomial rings $R[x]$ over $R$, and the completion of a Noetherian local ring.