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  "body": "\n## Summary\n\nThe tensor product $A \\otimes_R B$ of two [[algebras|DPBJ04JMH8]] over a commutative ring $R$ is the [[tensor product of modules|F0HQXQ0HCD]] with the multiplication $(a \\otimes b)(a' \\otimes b') = aa' \\otimes bb'$.\nFor commutative algebras it is the coproduct.\nA pair of algebra maps out of $A$ and $B$ to a commutative algebra is the same as one algebra map out of $A \\otimes_R B$.\nIt describes base change, such as $\\C \\otimes_\\R A$, and in geometry it gives the fibre product of affine schemes.\n\n## Explanations\n\n### L1 human\n\nThe tensor product of two algebras is the most general algebra built from them in which elements of the first commute with elements of the second.\nPolynomial rings show the idea: $R[x] \\otimes_R R[y]$ is $R[x, y]$.\nHere $x^2 \\otimes y$ corresponds to the monomial $x^2 y$.\nCombining two algebras this way can also split them up: $\\C \\otimes_\\R \\C$ is isomorphic to $\\C \\times \\C$.\n",
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