---
tag: 6B1PSZ42MC
kind: concept
name: Directly indecomposable algebra
msc: [08B25, 08A05]
sources:
  - url: https://en.wikipedia.org/wiki/Direct_product
---

## Summary

A directly indecomposable algebra is a nontrivial [[algebra|QJGPT2W0FT]] that is not isomorphic to a [[direct product|WJ73XYBNE4]] of two nontrivial algebras.
Every finite algebra is a direct product of finitely many directly indecomposable ones.
The notion is weaker than [[subdirect irreducibility|DASF939439]], which forbids also the more flexible subdirect decompositions.

## Explanations

### L1 human

Some algebras split into two independent parts, like $\Z/6\Z$, which behaves as $\Z/2\Z$ and $\Z/3\Z$ side by side.
A directly indecomposable algebra admits no such splitting.
Any algebra with a prime number of elements is directly indecomposable, because a product of two nontrivial algebras has a composite number of elements.
The ring $\Z$ is directly indecomposable too, even though it is not subdirectly irreducible.