---
tag: 6N268PHKQA
kind: concept
name: Factor congruence
msc: [08A30, 08B25]
specializes: [DCSXXT4SER]
sources:
  - url: https://en.wikipedia.org/wiki/Congruence_relation
---

## Summary

A factor congruence of an [[algebra|QJGPT2W0FT]] is a [[congruence|DCSXXT4SER]] that comes from splitting the algebra as a [[direct product|WJ73XYBNE4]] of two factors.
It is the kernel of the projection onto one factor.
Factor congruences describe all the ways an algebra decomposes into a product of two algebras.
An algebra is [[directly indecomposable|6B1PSZ42MC]] when its only factor congruences are the trivial ones.

## Explanations

### L1 human

If an algebra is a product $\mathbf{B} \times \mathbf{C}$, two pairs $(b, c)$ and $(b', c')$ can be compared by their first coordinates only.
The relation "same first coordinate" is a congruence, and it is a factor congruence.
Factor congruences let one detect such a product splitting from inside the algebra, without knowing $\mathbf{B}$ and $\mathbf{C}$ in advance.
For example, $\Z/6\Z$ splits as $\Z/2\Z \times \Z/3\Z$, and congruence modulo $2$ is the matching factor congruence.