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  "body": "\n## Summary\n\nThe tensor product $M \\otimes_R N$ of two modules over a [[commutative ring|0310REJJY8]] $R$ is the module that turns bilinear maps on $M \\times N$ into linear maps.\nIt is generated by the symbols $m \\otimes n$, which are additive in each variable and let scalars move between the two factors.\nIt generalises the [[tensor product of vector spaces|0BF2DE2SYK]], but over a ring it can be much smaller than expected, even zero.\nTensor products are used for change of rings, for [[flatness|73RYYEVDKA]], and for the [[tensor-hom adjunction|AVXARAXHKY]].\n\n## Explanations\n\n### L1 human\n\nThe tensor product combines two modules into one, in which you can multiply an element of the first by an element of the second.\nThe rules are those of a product: $(m + m') \\otimes n = m \\otimes n + m' \\otimes n$, and a scalar can be moved from one side to the other.\nSometimes everything cancels.\nFor example, $\\Z/2\\Z \\otimes_\\Z \\Z/3\\Z = 0$: since $3 \\cdot 1 = 1$ in $\\Z/2\\Z$, one has $a \\otimes b = 3a \\otimes b = a \\otimes 3b = a \\otimes 0 = 0$.\n",
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