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  "body": "\n## Definition\n\nLet $k$ be a [[field|04C5ZBG1WT]], let $d, n \\ge 1$, and let $a_1, \\ldots, a_n \\in \\Z^d$.\nFor $a = (c_1, \\ldots, c_d) \\in \\Z^d$, write $t^a = t_1^{c_1} \\cdots t_d^{c_d}$ in the Laurent polynomial ring $k[t_1^{\\pm 1}, \\ldots, t_d^{\\pm 1}]$.\nLet $\\varphi \\colon k[x_1, \\ldots, x_n] \\to k[t_1^{\\pm 1}, \\ldots, t_d^{\\pm 1}]$ be the $k$-algebra [[homomorphism|0VTH8KPCG2]] with $\\varphi(x_i) = t^{a_i}$.\nThe *toric ideal* of $A = (a_1, \\ldots, a_n)$ is the [[kernel|0JN0SQG2JH]] $I_A = \\ker \\varphi$.\n\n## Remarks\n\nThe ideal $I_A$ is [[prime|00MHHZ9SNE]], since $k[x_1, \\ldots, x_n]/I_A$ is isomorphic to a subring of a domain.\nLet $A$ also denote the $d \\times n$ integer matrix with columns $a_i$.\nThen $I_A$ is generated, as a $k$-vector space and hence as an ideal, by the binomials $x^u - x^v$ with $u, v \\in \\N^n$ and $Au = Av$.\n\nExample: $d = 2$, $n = 3$ and $a_1 = (1, 0)$, $a_2 = (1, 1)$, $a_3 = (1, 2)$, so $\\varphi(x_1) = t_1$, $\\varphi(x_2) = t_1 t_2$, $\\varphi(x_3) = t_1 t_2^2$.\nSince $\\varphi(x_1 x_3) = t_1^2 t_2^2 = \\varphi(x_2^2)$, the binomial $x_1 x_3 - x_2^2$ lies in $I_A$, and in fact $I_A = (x_1 x_3 - x_2^2)$.\n\nNon-example: if the characteristic of $k$ is not $2$, the binomial ideal $(x^2 - y^2)$ of $k[x, y]$ is not a toric ideal.\nIt is not prime, because $(x - y)(x + y) \\in (x^2 - y^2)$ while neither factor is.\n",
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