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  "body": "\n## Definition\n\nA division ring is a [[ring|0894J980RK]] $D$ with these two properties.\n\n1. $1 \\neq 0$ in $D$.\n2. Every nonzero element of $D$ is a [[unit|019NFGFPR5]]: for each $a \\neq 0$ in $D$, there is $b \\in D$ with $a b = b a = 1$.\n\n## Remarks\n\nA commutative division ring is a [[field|0QA8CHH8MG]].\nThe [[quaternions|0GE4DVQKR8]] are a division ring that is not commutative.\nBy [[Wedderburn's little theorem|0RRNVX1VKA]], every finite division ring is a field.\n\n## Explanations\n\n### L1 human\n\nIn a division ring, you can divide by every nonzero element, as in a field.\nBut $a b$ and $b a$ can be different.\nThe quaternions of Hamilton are the standard example.\n",
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