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  "body": "\n## Definition\n\nLet $R$ be an [[integral domain|09S3PFBEPM]].\n$R$ is a *GCD domain* if for all $a, b \\in R$ there is an element $d \\in R$ such that:\n\n1. $d \\mid a$ and $d \\mid b$;\n2. for every $c \\in R$ with $c \\mid a$ and $c \\mid b$, one has $c \\mid d$.\n\nSuch a $d$ is a [[greatest common divisor|0F7KNJ1ZYN]] of $a$ and $b$.\n\n## Remarks\n\nHere $x \\mid y$ means [[divisibility|04GNK7K4VP]] in $R$; a greatest common divisor is unique up to multiplication by a unit.\nExample: $\\Z$ is a GCD domain, and so is every [[unique factorisation domain|06YMXJMXYS]], where a gcd is read off from the factorizations.\nNon-example: in $\\Z[\\sqrt{-5}]$, with the norm $N(x + y\\sqrt{-5}) = x^2 + 5y^2$, the elements $6$ and $2(1 + \\sqrt{-5})$ have no gcd.\nIndeed $2$ and $1 + \\sqrt{-5}$ are common divisors, so the norm of a gcd $d$ would be divisible by $N(2) = 4$ and by $N(1 + \\sqrt{-5}) = 6$.\nAlso $N(d)$ would divide $N(6) = 36$ and $N(2 + 2\\sqrt{-5}) = 24$, so $N(d) = 12$.\nBut $x^2 + 5y^2 = 12$ has no integer solution, so clause 2 fails for every common divisor $d$.\nEquivalent form: any two elements have a least common multiple.\n",
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