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  "body": "\n## Summary\n\nA split exact sequence is a [[short exact sequence|KJKNJMGKZZ]] $0 \\to A \\to B \\to C \\to 0$ in which the middle object is the [[biproduct|BZEJKTQ76T]] $A \\oplus C$, compatibly with the two maps.\nThese are the trivial extensions: $B$ is just $A$ and $C$ placed side by side.\nBy the [[splitting lemma|7ZGQKWYKQS]], a short exact sequence splits as soon as one of its two maps has a one-sided inverse.\n\n## Explanations\n\n### L1 human\n\nEvery short exact sequence builds $B$ from a piece $A$ and a quotient $C$.\nIn a split sequence the gluing is trivial, and $B$ is simply the direct sum of $A$ and $C$.\nFor example, $0 \\to \\Z \\to \\Z \\oplus \\Z/2\\Z \\to \\Z/2\\Z \\to 0$ is split.\nThe sequence $0 \\to \\Z \\xrightarrow{2} \\Z \\to \\Z/2\\Z \\to 0$ is not, since $\\Z$ is not $\\Z \\oplus \\Z/2\\Z$.\nOver a field every short exact sequence of vector spaces splits, which is why linear algebra has no interesting extensions.\n",
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