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  "body": "\n## Statement\n\nThe [[integers|0C4JS9RF5M]] $\\mathbb{Z}$, with their addition $+$ and multiplication $\\cdot$, form a commutative ring with identity $1$.\nThat is, for all integers $a$, $b$ and $c$:\n\n1. $(a + b) + c = a + (b + c)$ and $a + b = b + a$;\n2. $a + 0 = a$, and there is an integer $-a$ with $a + (-a) = 0$;\n3. $(a b) c = a (b c)$ and $a b = b a$;\n4. $a \\cdot 1 = a$;\n5. $a (b + c) = a b + a c$.\n\n## Remarks\n\nThe ring $\\mathbb{Z}$ is not a field: for example, $2$ has no multiplicative inverse in $\\mathbb{Z}$.\nThe ring $\\mathbb{Z}$ has no zero divisors: if $a b = 0$, then $a = 0$ or $b = 0$. Thus it is an integral domain.\nHere \"ring of integers\" means the ring $\\mathbb{Z}$. In algebraic number theory, the same words have a more general meaning.\n\n## Explanations\n\n### L1 human\n\nThis statement lists the usual rules of computation with whole numbers.\nYou can add, subtract and multiply integers, and the result is always an integer.\nDivision is different: $1 / 2$ is not an integer.\n",
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