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  "body": "\n## Definition\n\nLet $R$ be a [[ring|0894J980RK]] and let $X \\subseteq R$ be a subset.\nThe left ideal generated by $X$ is the intersection of all the left [[ideals|0MYSA6MYNA]] of $R$ that contain $X$.\nThe notation for it is $(X)$.\nIt is the smallest left ideal of $R$ that contains $X$.\n\n## Remarks\n\nThe left ideal $(X)$ is the set of the finite sums $r_1 x_1 + \\dots + r_n x_n$ with $n \\ge 0$, $r_i \\in R$ and $x_i \\in X$.\nFor a finite set, one writes $(x_1, \\dots, x_n)$ for $(\\{x_1, \\dots, x_n\\})$.\nIn $\\mathbb{Z}$, the ideal $(4, 6)$ is $2\\mathbb{Z}$.\nAn ideal generated by one element is a [[principal ideal|00A2EDFT9Q]].\n\n## Explanations\n\n### L1 human\n\nStart with some elements of the ring.\nThe ideal that they generate contains all the sums of their multiples, and nothing else.\nIn the integers, the multiples of $4$ and $6$ give all the even numbers, because $2 = 2 \\cdot 4 - 1 \\cdot 6$.\n",
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