---
tag: 10TJ890QK0
kind: definition
name: Quotient algebra
msc: [08A30]
defines: N5KE7FH6CX
uses: [QJGPT2W0FT, DCSXXT4SER, 0011MTQMR6]
sources:
  - url: https://en.wikipedia.org/wiki/Quotient_(universal_algebra)
---

## Definition

Let $\mathbf{A}$ be an [[algebra|QJGPT2W0FT]] with universe $A$ and let $\theta$ be a [[congruence|DCSXXT4SER]] on $\mathbf{A}$.
For $a \in A$, let $a/\theta$ be the class of $a$, and let $A/\theta$ be the [[quotient set|0011MTQMR6]] of all classes.
The *quotient algebra* $\mathbf{A}/\theta$ has universe $A/\theta$ and the same signature as $\mathbf{A}$.
Each $n$-ary fundamental operation $f$ is interpreted by
$$f^{\mathbf{A}/\theta}(a_1/\theta, \dots, a_n/\theta) = f^{\mathbf{A}}(a_1, \dots, a_n)/\theta.$$

## Remarks

The operations are well defined because $\theta$ is compatible with each $f$.
The map $\nu_\theta \colon a \mapsto a/\theta$ is a surjective homomorphism $\mathbf{A} \to \mathbf{A}/\theta$, the natural map.

Example: for the ring $\Z$ and congruence modulo $n$, the quotient $\Z/\theta$ is the ring $\Z/n\Z$.
For instance $(3/\theta) \cdot (4/\theta) = 12/\theta = 2/\theta$ when $n = 5$.

Non-example: for an equivalence relation that is not compatible, the formula gives no operation.
Take $(\Z, +)$ with the classes "negative", "zero" and "positive".
Computing with $1$ and $-1$ gives the class of $0$, but computing with $2$ and $-1$ gives the class of $1$.

For a group $G$ and the congruence of a normal subgroup $N$, the quotient algebra is the [[quotient group|0KYX2N69J0]] $G/N$.