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  "body": "\n## Statement\n\nLet $R$ be a [[commutative ring|0GYGHKX834]].\nThen the [[polynomial ring|0Q9EZS90GY]] $R[x]$ is an [[integral domain|09S3PFBEPM]] if and only if $R$ is an integral domain.\n\n## Remarks\n\nIf $R$ is an integral domain and $f, g \\neq 0$, then the leading coefficient of $f g$ is not zero, so $f g \\neq 0$.\nConversely, $R$ is a subring of $R[x]$, and a subring of an integral domain is an integral domain.\nBy induction, the ring $R[x_1, \\dots, x_n]$ of [[polynomials in several variables|0CES91M79A]] over an integral domain is an integral domain.\n\n## Explanations\n\n### L1 human\n\nThe product of two nonzero polynomials over the integers is never zero.\nThe reason is that the leading coefficients multiply to a nonzero number.\nSo the polynomial ring keeps the integral domain property of its coefficients.\n",
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