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  "body": "\n## Definition\n\nLet $R$ be a [[commutative ring|0310REJJY8]] and let $M$, $N$ be [[$R$-modules|0NZ8JA25WK]].\nA map $\\beta \\colon M \\times N \\to P$ to an $R$-module $P$ is *$R$-bilinear* if it is $R$-[[linear|0NRYHJPQG2]] in each variable when the other is fixed.\nA *tensor product* of $M$ and $N$ is an $R$-module $M \\otimes_R N$ with an $R$-bilinear map $\\otimes \\colon M \\times N \\to M \\otimes_R N$, written $(m, n) \\mapsto m \\otimes n$.\nIt must satisfy this universal property.\nFor every $R$-module $P$ and every $R$-bilinear map $\\beta \\colon M \\times N \\to P$ there is exactly one $R$-linear map $\\varphi \\colon M \\otimes_R N \\to P$ with $\\varphi(m \\otimes n) = \\beta(m, n)$ for all $m \\in M$ and $n \\in N$.\n\n## Remarks\n\nA tensor product exists and is unique up to a unique isomorphism compatible with the maps $\\otimes$.\nOne construction is the [[free module|0DZ9DKAKZR]] on the set $M \\times N$ modulo the submodule generated by the elements $(m + m', n) - (m, n) - (m', n)$, $(m, n + n') - (m, n) - (m, n')$, $(rm, n) - r(m, n)$ and $(m, rn) - r(m, n)$.\nThe elements $m \\otimes n$ generate $M \\otimes_R N$, but a general element is a finite sum $\\sum_i m_i \\otimes n_i$.\nStandard isomorphisms are $R \\otimes_R N \\cong N$, $M \\otimes_R N \\cong N \\otimes_R M$, and $R^m \\otimes_R R^n \\cong R^{mn}$.\n\nExample: for integers $a, b \\ge 1$, $\\Z/a\\Z \\otimes_\\Z \\Z/b\\Z \\cong \\Z/d\\Z$ with $d = \\gcd(a, b)$.\nThe map $(x, y) \\mapsto xy \\bmod d$ is well defined and bilinear, so it gives a surjection onto $\\Z/d\\Z$; conversely $1 \\otimes 1$ generates and is killed by $a$ and $b$, hence by $d$.\n\nNon-example: the module $M \\oplus N$ with the map $(m, n) \\mapsto (m, n)$ is not a tensor product when $M$ and $N$ are nonzero, because this map is not bilinear.\nFor instance, with $M = N = \\Z$, it sends $(2, 1)$ to $(2, 1)$, while bilinearity would require $2 \\cdot (1, 1) = (2, 2)$.\n",
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