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.