Associative algebra

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Aassociative 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.

In this article we will also use the term K-algebra to mean
an associative algebra over the field K.
A standard first example of a K-algebra is a ring of square matrices over a field K,
with the usual matrix multiplication.

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A commutative algebra is an associative algebra that has a commutative multiplication,
or, equivalently, an associative algebra that is also a commutative ring.

In this article associative algebras are assumed to have a multiplicative identity, denoted 1; they are sometimes called unital associative algebras for clarification.

In some areas of mathematics this assumption is not made,
and we will call such structures non-unital associative algebras.
We will also assume that all rings are unital, and all ring homomorphisms are unital.

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