---
tag: NY25HXA478
kind: concept
name: Semisimple ring
msc: [16D60, 16K99, 16P20]
specializes: [0B1261B5ED]
sources:
  - url: https://en.wikipedia.org/wiki/Semisimple_module
---

## Summary

A semisimple ring is a ring that is a [[semisimple module|N3MRHDKDGC]] over itself: it is a direct sum of minimal left ideals.
Every module over a semisimple ring is semisimple, so every submodule has a complement and every module is projective and injective.
The Artin–Wedderburn theorem describes them completely as finite products of matrix rings over division rings.
Group algebras of finite groups over fields of characteristic zero are the main examples, by Maschke's theorem.

## Explanations

### L1 human

In linear algebra over a field, every subspace has a complementary subspace.
A semisimple ring is a ring whose modules behave in the same way: each submodule splits off as a direct summand.
So modules over a semisimple ring break into simple pieces, like vector spaces break into lines.
The ring $\R \times \R$ and the ring of $2 \times 2$ complex matrices are semisimple, while $\Z$ is not, since $2\Z$ has no complement in $\Z$.