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  "body": "\n## Definition\n\nLet $R$ and $S$ be [[rings|0894J980RK]].\nA ring homomorphism from $R$ to $S$ is a map $f \\colon R \\to S$ with these three properties.\n\n1. $f(a + b) = f(a) + f(b)$ for all $a, b \\in R$.\n2. $f(a b) = f(a) f(b)$ for all $a, b \\in R$.\n3. $f(1) = 1$.\n\n## Remarks\n\nA ring homomorphism also satisfies $f(0) = 0$ and $f(-a) = -f(a)$.\nFor each integer $n \\ge 1$, the map $\\mathbb{Z} \\to \\mathbb{Z}/n\\mathbb{Z}$ that sends $a$ to its class is a ring homomorphism.\nThe map $\\mathbb{Z} \\to \\mathbb{Z}$, $a \\mapsto 2a$, preserves sums but is not a ring homomorphism.\nA bijective ring homomorphism is a [[ring isomorphism|0B6QC0C0G1]].\n\n## Explanations\n\n### L1 human\n\nA ring homomorphism translates one ring into another and keeps the arithmetic.\nIt does not matter if you add or multiply first and then apply $f$, or apply $f$ first.\nReduction modulo $n$ is the main example.\n",
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