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  "body": "\n## Summary\n\nA separable algebra over a commutative ring $K$ is an [[associative algebra|3CTW1P4HBP]] $A$ whose multiplication map $A \\otimes_K A^{\\mathrm{op}} \\to A$ splits as a map of bimodules.\nOver a field, for finite-dimensional $A$, it means that $A$ stays [[semisimple|NY25HXA478]] after every extension of the base field.\nIt generalizes the notion of a [[separable field extension|0WCX7NB4N7]] to noncommutative algebras, and it is the starting point of the theory of Azumaya algebras.\n\n## Explanations\n\n### L1 human\n\nA semisimple algebra can lose its semisimplicity when the base field is enlarged, in the same way as an irreducible polynomial can acquire a repeated root.\nSeparable algebras are the ones for which this never happens.\nFor example, $\\C$ is separable over $\\R$: after tensoring with $\\C$ it becomes $\\C \\times \\C$, which is still semisimple.\nOver fields of characteristic $0$, and over finite fields, every finite-dimensional semisimple algebra is separable.\n",
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