---
tag: 3CTW1P4HBP
kind: concept
name: Associative algebra
msc: [16B99, 16S99]
specializes: [0B1261B5ED]
sources:
  - url: https://en.wikipedia.org/wiki/Associative_algebra
same_as:
  wikidata: Q744960
  wikipedia: Associative algebra
---

## Summary

An associative algebra over a commutative ring $K$ is a [[ring|0B1261B5ED]] whose elements can also be multiplied by scalars from $K$.
The scalar product is compatible with the ring product.
Matrix algebras, polynomial algebras, group algebras and endomorphism algebras of vector spaces are the standard examples.
The notion puts ring theory and linear algebra in one frame, so that dimension and bases are available when $K$ is a field.
A [[commutative algebra over a ring|DPBJ04JMH8]] is the special case where the ring product is commutative.

## Explanations

### L1 human

An algebra is a vector space, or more generally a module, in which vectors can also be multiplied with each other.
The multiplication is associative and works well with scaling: scaling a product is the same as scaling one factor.
The $n \times n$ real matrices form an algebra over $\R$: they can be added, multiplied, and scaled by real numbers.
The complex numbers are a $2$-dimensional algebra over $\R$, with basis $1, i$ and the rule $i^2 = -1$.