---
tag: 8F6EHDVJ62
kind: concept
name: Primitive ring
msc: [16D60]
specializes: [Z9TJSZZYG1]
sources:
  - url: https://en.wikipedia.org/wiki/Primitive_ring
same_as:
  wikidata: Q2895708
  wikipedia: Primitive ring
---

## Summary

A left primitive ring is a ring that has a faithful simple left module.
By the [[Jacobson density theorem|EK0T7FRKCY]], such a ring acts densely on a vector space over a division ring.
So it is close to a full ring of linear maps.
Every left primitive ring is [[prime|Z9TJSZZYG1]], and a commutative ring is primitive exactly when it is a field.
Primitive rings are the building blocks of Jacobson's structure theory: every [[semiprimitive ring|1B2DBKV1C9]] is a subdirect product of primitive rings.

## Explanations

### L1 human

A module is a set on which the ring acts by something like linear maps.
A ring is primitive if it can act on one irreducible module without losing information, that is, with no nonzero element acting as zero.
The typical example is the ring of all linear maps of a vector space $V$, acting on $V$ itself.
The integers are not primitive: each simple $\Z$-module $\Z/p\Z$ is killed by $p$.