# Associative Algebra

[https://en.wikipedia.org/wiki/Associative\_algebra](https://en.wikipedia.org/wiki/Associative_algebra)

An **associative algebra** is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a _scalar multiplication_ by elements in some _field_.

The addition and multiplication operations together give `A` the structure of a _ring_; the addition and scalar multiplication operations together give `A` the structure of a _vector space over `K`_.

The term _K-algebra_ is used to mean an _associative algebra over the field `K`_.

The standard example of a K-algebra is a ring of square matrices over a field `K`, with the usual matrix multiplication.

Associative algebras are assumed to have a _multiplicative identity_, denoted by `1` (they are sometimes called _unital associative algebras_), and that all _rings_, as well as all _ring homomorphisms_, are _unital_ .

A **commutative algebra** is an associative algebra that has a _commutative multiplication_, or, equivalently, an associative algebra that is also a _commutative ring_.
