---
tag: 1TZSV19D85
kind: definition
name: Associative algebra
msc: [16B99, 16S99]
defines: 3CTW1P4HBP
uses: [0894J980RK, 0GYGHKX834, 09VDS2W1E8]
sources:
  - url: https://en.wikipedia.org/wiki/Associative_algebra
---

## Definition

Let $K$ be a [[commutative ring|0GYGHKX834]].
Let $A$ be a [[ring|0894J980RK]] whose additive group carries a [[$K$-module|09VDS2W1E8]] structure.
Then $A$ is an *associative algebra over $K$*, or *$K$-algebra*, if
$$k(ab) = (ka)b = a(kb)$$
for all $k \in K$ and $a, b \in A$.

## Remarks

Equivalently, a $K$-algebra is a ring $A$ with a ring homomorphism $\eta \colon K \to A$ whose image lies in the center of $A$; the scalar product is then $ka = \eta(k)a$.
Example: for a field $F$, the matrix ring $M_n(F)$ is an $F$-algebra with the entrywise scalar product.
The check: $k(AB)$, $(kA)B$ and $A(kB)$ all have $(i,j)$ entry $k \sum_l a_{il} b_{lj}$, because $k$ commutes with the entries.
Non-example: the quaternions $\mathbb{H}$ with left multiplication by elements of $\C \subset \mathbb{H}$ are not a $\C$-algebra.
The condition $k(ab) = a(kb)$ fails for the scalar $i$ with $a = j$ and $b = 1$, since $i(j \cdot 1) = ij$ while $j(i \cdot 1) = ji = -ij \ne ij$.
They are an algebra over $\R$, because real numbers commute with all quaternions.
Every ring is a $\Z$-algebra in exactly one way.
Some authors do not require a unit element in $A$; this note assumes one.