---
tag: 8831C5ARPB
kind: definition
name: Faithfully flat module
msc: [13C11, 13B40]
defines: C31PYYPRGC
uses: [0310REJJY8, 0NZ8JA25WK, 73RYYEVDKA, F0HQXQ0HCD, JB7ZBFBA9C, 0PV74TR5FP]
sources:
  - url: https://en.wikipedia.org/wiki/Flat_module
---

## Definition

Let $R$ be a [[commutative ring|0310REJJY8]] and let $M$ be an [[$R$-module|0NZ8JA25WK]].
The module $M$ is *faithfully flat* if both conditions hold:

1. $M$ is [[flat|73RYYEVDKA]].
2. For every $R$-module $N$, if $N \otimes_R M = 0$, then $N = 0$.

## Remarks

The following are equivalent for an $R$-module $M$:

1. $M$ is faithfully flat.
2. A sequence $N' \to N \to N''$ of $R$-modules is [[exact|JB7ZBFBA9C]] if and only if $N' \otimes_R M \to N \otimes_R M \to N'' \otimes_R M$ is exact.
3. $M$ is flat and $\mathfrak{m} M \ne M$ for every [[maximal ideal|0PV74TR5FP]] $\mathfrak{m}$ of $R$.

Example: a nonzero free module $M = R^{(I)}$ is faithfully flat.
It is flat, and $N \otimes_R R^{(I)} \cong N^{(I)}$, which is zero only when $N = 0$, since $I$ is not empty.

Non-example: $\Q$ is a flat $\Z$-module, so clause 1 holds.
But $\Z/2\Z \otimes_\Z \Q = 0$ while $\Z/2\Z \ne 0$, since $1 \otimes q = 2 \otimes \frac{q}{2} = 0 \otimes \frac{q}{2} = 0$; so clause 2 fails.