---
tag: M0JDQR0V5S
kind: concept
name: Primitive ideal
msc: [16D60, 16N20]
specializes: [E8K2MN5V1E]
sources:
  - url: https://en.wikipedia.org/wiki/Primitive_ideal
same_as:
  wikidata: Q7243577
  wikipedia: Primitive ideal
---

## Summary

A left primitive ideal of a ring is the annihilator of a simple left module.
Equivalently, it is a two-sided ideal $P$ such that $R/P$ is a [[primitive ring|8F6EHDVJ62]].
The [[Jacobson radical|XA0DCNH7WY]] of a ring is the intersection of its primitive ideals.
Every primitive ideal is [[prime|E8K2MN5V1E]], and in a commutative ring the primitive ideals are the maximal ideals.

## Explanations

### L1 human

Each simple module of a ring is killed by some elements of the ring.
The set of these elements is a primitive ideal.
In $\Z$ the simple modules are $\Z/p\Z$, and the primitive ideals are the ideals $p\Z$ for primes $p$.
Primitive ideals play for noncommutative rings the role that maximal ideals play for commutative rings.