---
tag: 88WN3AD2VQ
kind: concept
name: Subdirect product
msc: [08B26]
sources:
  - url: https://en.wikipedia.org/wiki/Subdirect_product
same_as:
  wikidata: Q1691827
  wikipedia: Subdirect product
---

## Summary

A subdirect product of a family of [[algebras|QJGPT2W0FT]] is a [[subalgebra|7CHG6SD365]] of their [[direct product|WJ73XYBNE4]] that still projects onto each factor.
It can be much smaller than the full product, yet it remembers each factor completely.
By [[Birkhoff's subdirect representation theorem|CZF5696TPY]], every algebra is isomorphic to a subdirect product of [[subdirectly irreducible algebras|DASF939439]].

## Explanations

### L1 human

The full product of two algebras contains every pair of elements.
A subdirect product keeps only some pairs, but enough so that every element of each factor still appears in some kept pair.
For example, the diagonal $\set{(x, x)}$ in $\mathbf{A} \times \mathbf{A}$ is a subdirect product of two copies of $\mathbf{A}$.
The integers embed in the product of all $\Z/p\Z$ for primes $p$ by sending $n$ to its list of remainders, and this is a subdirect product too.