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  "body": "\n## Definition\n\nLet $f \\colon R \\to S$ be a [[ring homomorphism|0GGE0Z0JQ1]].\nThe kernel of $f$ is the set\n$$\\ker f = \\{a \\in R : f(a) = 0\\}.$$\n\n## Remarks\n\nThe kernel of $f$ is a [[two-sided ideal|0DDA6CY7GJ]] of $R$.\nThe kernel of the reduction map $\\mathbb{Z} \\to \\mathbb{Z}/n\\mathbb{Z}$ is $n\\mathbb{Z}$.\nThe map $f$ is injective if and only if $\\ker f = \\{0\\}$, by the [[kernel criterion|0MMCB8RXGA]].\n\n## Explanations\n\n### L1 human\n\nThe kernel collects all the elements that $f$ sends to $0$.\nFor reduction modulo $5$, the kernel is the set of the multiples of $5$.\n",
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