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  "body": "\n## Definition\n\nA principal ideal domain is an [[integral domain|09S3PFBEPM]] $R$ in which every ideal is [[principal|00A2EDFT9Q]].\nThat is, for every ideal $I$ of $R$, there is $a \\in R$ with $I = (a)$.\n\n## Remarks\n\nEvery [[Euclidean domain is a principal ideal domain|0QKVE1GD9A]]. So $\\mathbb{Z}$, $K[x]$ for a field $K$, and $\\mathbb{Z}[i]$ are principal ideal domains.\nThe ring $\\mathbb{Z}[x]$ is [[not a principal ideal domain|06E3AG9TM2]].\nEvery principal ideal domain is a [[unique factorization domain|06YMXJMXYS]].\nThe ring $\\mathbb{Z}[\\tfrac{1 + \\sqrt{-19}}{2}]$ is a principal ideal domain that is not a Euclidean domain.\n\n## Explanations\n\n### L1 human\n\nIn a principal ideal domain, every ideal is the set of the multiples of one element.\nFor the integers, every ideal is $n\\mathbb{Z}$ for some $n$.\nThis makes divisibility in a principal ideal domain behave much like divisibility of integers.\n",
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