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  "body": "\n## Definition\n\nLet $K$ be a commutative ring and let $A$ be an [[associative algebra|3CTW1P4HBP]] over $K$.\nLet $A^{\\mathrm{op}}$ be the [[opposite ring|M8Q24XB0AF]] of $A$ and let $A^e = A \\otimes_K A^{\\mathrm{op}}$ be the [[tensor product of algebras|1735VGRT2P]].\nAn $A$-[[bimodule|EM7FHAFD3N]] is the same as a left $A^e$-module, and $A$ itself is one.\nLet $\\mu \\colon A^e \\to A$ be the map $a \\otimes b \\mapsto ab$, which is a homomorphism of $A^e$-modules.\nThe algebra $A$ is *separable* over $K$ if there is a homomorphism of $A^e$-modules $s \\colon A \\to A^e$ with $\\mu \\circ s = \\id_A$.\n\n## Remarks\n\nEquivalently, $A$ is a projective $A^e$-module.\nEquivalently, there is a *separability idempotent*: an element $e = \\sum_i x_i \\otimes y_i$ of $A^e$ with $\\sum_i x_i y_i = 1$ and $\\sum_i a x_i \\otimes y_i = \\sum_i x_i \\otimes y_i a$ for all $a \\in A$; take $e = s(1)$.\n\nExample: $A = M_n(K)$ is separable over $K$.\nWith matrix units $E_{ij}$, the element $e = \\sum_{i} E_{i1} \\otimes E_{1i}$ satisfies $\\sum_i E_{i1} E_{1i} = \\sum_i E_{ii} = 1$, and for all $k, l$ the sums $\\sum_i E_{kl} E_{i1} \\otimes E_{1i}$ and $\\sum_i E_{i1} \\otimes E_{1i} E_{kl}$ both equal $E_{k1} \\otimes E_{1l}$.\n\nNon-example: let $K = \\F_p(t)$ and $A = K(t^{1/p})$, a purely inseparable field extension of degree $p$.\nThen $A \\otimes_K A \\cong A[x]/(x - t^{1/p})^p$ has nonzero nilpotent elements, so it is not semisimple, and $A$ is not separable over $K$.\n\nWhen $K$ is a field, a finite-dimensional algebra $A$ is separable if and only if $A \\otimes_K L$ is [[semisimple|NY25HXA478]] for every field extension $L$ of $K$.\nEquivalently, $A$ is semisimple and the center of each simple factor of $A$ is a [[separable field extension|0WCX7NB4N7]] of $K$.\nA finite field extension $L/K$ is a separable algebra exactly when it is a separable extension.\n",
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