---
tag: 5AFH94P4S3
kind: concept
name: Extension and contraction of ideals
msc: [13A15, 13B02]
sources:
  - url: https://en.wikipedia.org/wiki/Ideal_(ring_theory)
---

## Summary

A [[ring homomorphism|0GGE0Z0JQ1]] $f \colon A \to B$ 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|06HC6HK0Z8]] 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.

## Explanations

### L1 human

Think of the inclusion $\Z \subseteq \Z[i]$ of the integers in the [[Gaussian integers|0B268GVX95]].
The ideal $5\Z$ extends to the ideal $5\Z[i]$, and since $5 = (2 + i)(2 - i)$ this ideal is no longer prime.
Going the other way, the prime ideal $(2 + i)$ of $\Z[i]$ contracts to $(2+i) \cap \Z = 5\Z$.
Extension pushes ideals up to the bigger ring, and contraction pulls them back down.