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  "body": "\n## Summary\n\nA [[ring homomorphism|0GGE0Z0JQ1]] $f \\colon A \\to B$ of commutative rings moves ideals in both directions.\nAn ideal of $A$ extends to the ideal of $B$ generated by its image, and an ideal of $B$ contracts to its preimage in $A$.\nContraction keeps [[prime ideals|06HC6HK0Z8]] prime, while extension in general does not.\nThese two operations describe how ideals behave under localization, under integral extensions and in number fields, where they describe how primes split.\n\n## Explanations\n\n### L1 human\n\nThink of the inclusion $\\Z \\subseteq \\Z[i]$ of the integers in the [[Gaussian integers|0B268GVX95]].\nThe ideal $5\\Z$ extends to the ideal $5\\Z[i]$, and since $5 = (2 + i)(2 - i)$ this ideal is no longer prime.\nGoing the other way, the prime ideal $(2 + i)$ of $\\Z[i]$ contracts to $(2+i) \\cap \\Z = 5\\Z$.\nExtension pushes ideals up to the bigger ring, and contraction pulls them back down.\n",
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