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  "body": "\n## Definition\n\nLet $R$ be an [[integral domain|09S3PFBEPM]] and let $a, b \\in R$.\nAn element $d \\in R$ is a greatest common divisor of $a$ and $b$ if it has these two properties.\n\n1. $d \\mid a$ and $d \\mid b$.\n2. For every $c \\in R$ with $c \\mid a$ and $c \\mid b$, $c \\mid d$.\n\n## Remarks\n\nIf $d$ and $d'$ are two greatest common divisors of $a$ and $b$, then $d$ and $d'$ are [[associates|053E86NWVK]].\nIn $\\mathbb{Z}$, the greatest common divisors of $12$ and $18$ are $6$ and $-6$.\nA greatest common divisor need not exist. A GCD domain is an integral domain in which every pair of elements has one.\nIn a [[principal ideal domain|0RZYDVZ475]], $d$ is a greatest common divisor of $a$ and $b$ if and only if $(a, b) = (d)$.\n\n## Explanations\n\n### L1 human\n\nIn a general ring, \"greatest\" does not refer to size.\nIt means the largest for divisibility: every other common divisor divides it.\nFor the integers, this gives the usual gcd, up to the sign.\n",
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