---
tag: 5M223XDM43
kind: concept
name: Essential extension
msc: [08B30]
sources:
  - url: https://en.wikipedia.org/wiki/Essential_extension
---

## Summary

An essential extension of an algebra $A$ is a larger algebra $B$ containing $A$ that cannot collapse anything without collapsing part of $A$.
Every nontrivial [[congruence|DCSXXT4SER]] of $B$ already identifies two distinct elements of $A$.
Essential extensions are the building blocks of injective hulls: in good cases, the injective hull of $A$ is a maximal essential extension of $A$.

## Explanations

### L1 human

Think of $A$ as sitting inside $B$ and holding it together.
If you glue together some elements of $B$, you are forced to glue together some elements of $A$ as well.
For example, the integers sit essentially inside the rationals as additive groups.
Any nonzero subgroup of $\Q$ contains some fraction $p/q \ne 0$, and then it contains the nonzero integer $p = q \cdot (p/q)$.