---
tag: 4B66270PGS
kind: definition
name: Essential extension
msc: [08B30]
defines: 5M223XDM43
uses: [DCSXXT4SER, KBXSZC6BV1]
sources:
  - url: https://en.wikipedia.org/wiki/Essential_extension
---

## Definition

Fix a signature, let $B$ be an algebra and let $A$ be a [[subalgebra|7CHG6SD365]] of $B$.
The algebra $B$ is an *essential extension* of $A$ if every [[congruence|DCSXXT4SER]] $\theta$ of $B$ with $\theta \cap (A \times A) = \Delta_A$ is the trivial congruence $\Delta_B$.
Here $\Delta_X = \set{(x, x) : x \in X}$.

## Remarks

Equivalently, every [[homomorphism|KBXSZC6BV1]] $\varphi$ from $B$ whose restriction to $A$ is injective is itself injective.
An embedding $e \colon A \to B$ is called essential when $B$ is an essential extension of $e(A)$.
For modules over a ring, congruences correspond to submodules, and the condition says that every nonzero submodule of $B$ meets $A$ in a nonzero element.

Example: $\Q$ is an essential extension of $\Z$ as abelian groups.
Congruences of $\Q$ correspond to subgroups $N$, and $\theta \cap (\Z \times \Z) = \Delta_{\Z}$ means $N \cap \Z = \set{0}$.
If $N$ contains $p/q \ne 0$, then it contains the nonzero integer $p$, so $N \cap \Z = \set{0}$ forces $N = \set{0}$.

Non-example: $\Z \times \Z$ is not an essential extension of $\Z \times \set{0}$ as abelian groups.
The subgroup $N = \set{0} \times \Z$ meets $\Z \times \set{0}$ only in $0$, but $N \ne \set{0}$.