= Global dimension zero and split exact sequences
{title2=$\operatorname{gldim}R=0\iff\text{every short exact sequence splits}$}
If <global dimension> is zero, every <module> is a <projective module>. Each <short exact sequence> splits because its quotient is projective. Conversely, splitting every <short exact sequence> makes each <module> a direct summand of a <free module> and therefore a <projective module>. Splitting also makes every <module> an <injective module>: for $A\hookrightarrow B$ choose a retraction $B\to A$ and compose it with any prescribed map out of $A$.
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