Extension and contraction of ideals

Summary

A ring homomorphism $f \colon A \to B$ of commutative rings moves ideals in both directions. An ideal of $A$ extends to the ideal of $B$ generated by its image, and an ideal of $B$ contracts to its preimage in $A$. Contraction keeps prime ideals prime, while extension in general does not. These two operations describe how ideals behave under localization, under integral extensions and in number fields, where they describe how primes split.