= Projective modules are flat
{title2=$P\text{ projective}\implies P\text{ flat}$}
A <projective module> is a <direct summand> of a <free module>. <Free modules> are flat because tensoring with a <free module> produces a <direct sum> of copies of the original module and map. If $F=P\oplus Q$ is flat, tensoring an <injection> with $F$ is the <direct sum> of its tensor maps with $P$ and $Q$, so each component is <injective>. Applying this to a <free module> containing $P$ proves that $P$ is <flat>.
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