[[K-algebra]]
# $\mathbb{K}$-monoid

A **$\mathbb{K}$-monoid** is an [[K-algebra]] for which the bilinear product possesses a two-sided identity $1$ and is associative #m/def/falg 

- $1 a = a 1 = a$ for all $a \in A$
- $(ab)c = a(bc)$ for all $a,b,c \in A$

Hence it is also called a **unital associative algebra** over $\mathbb{K}$.
In other words, a $\mathbb{K}$-monoid is a field action on a [[ring]].
An algebra homomorphism which preserves the identity is a [[Algebra homomorphism|unital algebra homomorphism]] or **$\mathbb{K}$-monoid homomorphism**.
A generalization is an [[R-monoid]].

## Further terminology

 - [[Algebraic element]]

## Examples

- [[Matrix algebra over a field]]
- [[Complex number]]
- [[Quaternion]] (non-commutative)
- [[Endomorphism ring]]
- [[Tensor algebra]]
- [[Clifford algebra]]
- [[Extension field as a unital associative algebra]]

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#state/tidy | #lang/en | #SemBr