Direct summands of flat modules are flat
= Direct summands of flat modules are flat
{title2=$F=P\oplus Q\text{ flat}\implies P,Q\text{ flat}$}
If $F=P\oplus Q$ is a <flat module>, then $P$ and $Q$ are <flat>. For an <injection> $u:U\hookrightarrow V$, distributivity of the <tensor product of modules> identifies $u\otimes F$ with $(u\otimes P)\oplus(u\otimes Q)$. This map is <injective> by flatness of $F$. Each component is therefore <injective>, proving flatness of both summands.