---
tag: 1B2DBKV1C9
kind: concept
name: Semiprimitive ring
msc: [16N20]
specializes: [5F0YMCV50R]
sources:
  - url: https://en.wikipedia.org/wiki/Semiprimitive_ring
same_as:
  wikidata: Q7449622
  wikipedia: Semiprimitive ring
---

## Summary

A semiprimitive ring is a ring whose [[Jacobson radical|XA0DCNH7WY]] is zero.
Equivalently, it is a subdirect product of [[primitive rings|8F6EHDVJ62]], or it has a faithful semisimple module.
Every semiprimitive ring is [[semiprime|5F0YMCV50R]], and a left Artinian ring is semiprimitive exactly when it is a [[semisimple ring|NY25HXA478]].
The ring $R/J(R)$ is always semiprimitive, which is why the Jacobson radical is the first thing one removes in the structure theory of rings.

## Explanations

### L1 human

The Jacobson radical is the part of a ring that every simple module ignores.
A semiprimitive ring has no such part: each nonzero element acts nontrivially on some simple module.
The integers are semiprimitive, because no nonzero integer lies in every ideal $p\Z$.
The ring $\Z/4\Z$ is not, because its only maximal ideal is $2\Z/4\Z \ne 0$.