## Tag `00HB`

Chapter 10: Commutative Algebra > Section 10.38: Flat modules and flat ring maps

Definition 10.38.1. Let $R$ be a ring.

- An $R$-module $M$ is called
flatif whenever $N_1 \to N_2 \to N_3$ is an exact sequence of $R$-modules the sequence $M \otimes_R N_1 \to M \otimes_R N_2 \to M \otimes_R N_3$ is exact as well.- An $R$-module $M$ is called
faithfully flatif the complex of $R$-modules $N_1 \to N_2 \to N_3$ is exact if and only if the sequence $M \otimes_R N_1 \to M \otimes_R N_2 \to M \otimes_R N_3$ is exact.- A ring map $R \to S$ is called
flatif $S$ is flat as an $R$-module.- A ring map $R \to S$ is called
faithfully flatif $S$ is faithfully flat as an $R$-module.

The code snippet corresponding to this tag is a part of the file `algebra.tex` and is located in lines 8299–8317 (see updates for more information).

```
\begin{definition}
\label{definition-flat}
Let $R$ be a ring.
\begin{enumerate}
\item An $R$-module $M$ is called {\it flat} if whenever
$N_1 \to N_2 \to N_3$ is an exact sequence of $R$-modules
the sequence $M \otimes_R N_1 \to M \otimes_R N_2 \to M \otimes_R N_3$
is exact as well.
\item An $R$-module $M$ is called {\it faithfully flat} if the
complex of $R$-modules
$N_1 \to N_2 \to N_3$ is exact if and only if
the sequence $M \otimes_R N_1 \to M \otimes_R N_2 \to M \otimes_R N_3$
is exact.
\item A ring map $R \to S$ is called {\it flat} if
$S$ is flat as an $R$-module.
\item A ring map $R \to S$ is called {\it faithfully flat} if
$S$ is faithfully flat as an $R$-module.
\end{enumerate}
\end{definition}
```

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