---
tag: 62M0NC3E2T
kind: concept
name: Simple ring
msc: [16D60, 16D25]
specializes: [0B1261B5ED]
sources:
  - url: https://en.wikipedia.org/wiki/Simple_ring
same_as:
  wikidata: Q2326800
  wikipedia: Simple ring
---

## Summary

A simple ring is a nonzero ring whose only [[two-sided ideals|0CD87SFM2N]] are $0$ and the whole ring.
It cannot be broken into smaller rings by passing to quotients.
Fields, [[division rings|0EA5BTA0CX]] and full [[matrix rings|0FHVPRAE60]] over them are simple.
By Wedderburn's theorem, the simple Artinian rings are exactly the matrix rings over division rings.

## Explanations

### L1 human

For a commutative ring, having no ideals except $0$ and itself means being a field.
For noncommutative rings, one asks this only of two-sided ideals, and many more rings qualify.
The ring of $2 \times 2$ real matrices is simple, although it has zero divisors and one-sided ideals.
Simple rings play the role among rings that simple groups play among groups.