---
tag: 5F0YMCV50R
kind: concept
name: Semiprime ring
msc: [16N60]
sources:
  - url: https://en.wikipedia.org/wiki/Semiprime_ring
same_as:
  wikidata: Q139893873
  wikipedia: Semiprime ring
---

## Summary

A semiprime ring is a ring in which $aRa = 0$ forces $a = 0$.
Equivalently, it has no nonzero nilpotent two-sided ideals, or its [[prime radical|WF117EJ2HF]] is zero.
Semiprime rings are the subdirect products of [[prime rings|Z9TJSZZYG1]], and a commutative ring is semiprime exactly when it is [[reduced|SZHY975FFX]].
They form the natural setting of Goldie's theorem and of much of the theory of polynomial identity rings.

## Explanations

### L1 human

A reduced commutative ring has no nonzero nilpotent elements.
In noncommutative rings this is too strong, since matrix rings have nilpotent matrices.
A semiprime ring instead has no nonzero ideal whose elements all multiply to zero in long enough products.
For example $M_2(\Q)$ is semiprime, while the ring of upper triangular $2 \times 2$ matrices is not, because its strictly upper triangular matrices form an ideal with square zero.