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  "body": "\n## Definition\n\nLet $R$ be a [[ring|0894J980RK]].\nA left ideal of $R$ is a subset $I \\subseteq R$ with these properties.\n\n1. $I$ is a subgroup of the additive [[group|G5S1M9]] $(R, +)$.\n2. For all $r \\in R$ and $a \\in I$, $r a \\in I$.\n\nIf $R$ is a [[commutative ring|0GYGHKX834]], a left ideal is simply called an ideal.\n\n## Remarks\n\nIn $\\mathbb{Z}$, the set $n\\mathbb{Z}$ of the multiples of $n$ is an ideal for every integer $n$.\nEvery ring $R$ has the ideals $\\{0\\}$ and $R$.\nAn ideal contains $1$ if and only if it is all of $R$.\nIn a noncommutative ring, one also uses right ideals and [[two-sided ideals|0DDA6CY7GJ]].\n\n## Explanations\n\n### L1 human\n\nAn ideal is a set that absorbs multiplication.\nThe even integers are an example: an even number times any integer is even.\nIdeals play the role in rings that normal subgroups play in groups.\n",
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