---
tag: TMTPWVETQK
kind: concept
name: Primary ideal
msc: ["13A15"]
specializes: [096Z9MP2ME]
sources:
  - url: https://en.wikipedia.org/wiki/Primary_ideal
same_as:
  wikidata: Q1142699
  wikipedia: Primary ideal
---

## Summary

A primary ideal of a commutative ring is a proper ideal $Q$ with this property.
Whenever a product $ab$ lies in $Q$ and $a$ does not, some power of $b$ lies in $Q$.
Primary ideals generalise powers of prime numbers: in $\Z$ the nonzero primary ideals are the ideals $(p^n)$ for primes $p$.
The [[radical|437YRCNYRJ]] of a primary ideal is a [[prime ideal|00MHHZ9SNE]].
Primary ideals are the building blocks of [[primary decomposition|ZD34THCXQ0]].

## Explanations

### L1 human

In the integers, every number is a product of prime powers, and the ideals $(p^n)$ are the primary ideals.
A primary ideal is "concentrated at one prime": its radical is a single prime ideal, as $(p^n)$ has radical $(p)$.
For example, $(8) = (2^3)$ is primary in $\Z$, but $(6)$ is not, since $6$ involves two different primes.