An **associative algebra** is a vector space which also allows the multiplication of vectors in a distributive and associative manner.
### Definition

An associative algebra *A* over a field *K* is defined to be a vector space over *K* together with a *K*-bilinear multiplication *A* x *A* `->` *A* (where the image of (*x*,*y*) is written as *xy*) such that the associativity law holds:
*A* contains an identity element, i.e. an element 1 such that 1*x* = *x*1 = *x* for all *x* in *K*, then we call *A* an *associative algebra with one* or a *unitary associative algebra*. Such an algebra is a ring.
### Examples

### Algebra homomorphisms

### Generalizations

- (
*x y*)*z*=*x*(*y z*) for all*x*,*y*and*z*in*A*.

- (
*x*+*y*)*z*=*x z*+*y z*for all*x*,*y*,*z*in*A*, -
*x*(*y*+*z*) =*x y*+*x z*for all*x*,*y*,*z*in*A*, - α (
*x y*) = (α*x*)*y*=*x*(α*y*) for all*x*,*y*in*A*and α in*K*.

The *dimension* of the associative algebra *A* over the field *K* is its dimension as a *K*-vector space.

- The square
*n*-by-*n*matrices with entries from the field*K*form a unitary associative algebra over*K*. - The complex numbers form a 2-dimensional unitary associative algebra over the real numbers
- The quaternions form a 4-dimensional unitary associative algebra over the reals and a 2-dimensional unitary associative algebra over the complex numbers.
- The polynomials with real coefficients form a unitary associative algebra over the reals.
- Given any Banach space
*X*, the continuous linear operators*A*:*X*`->`*X*form a unitary associative algebra; this is in fact a Banach algebra. - Given any topological space
*X*, the continuous real- (or complex-) valued functions on*X*form a real (or complex) unitary associative algebra. - An example of a non-unitary associative algebra is given by the functions
*f*:**R**`->`**R**whose limit for*x*→∞ is zero.

If *A* and *B* are associative algebras over the same field *K*, an *algebra homomorphism* *f* : *A* `->` *B* is a *K*-linear map which is also multiplicative in the sense that *f*(*xy*) = *f*(*x*) *f*(*y*) for all *x*, *y* in *A*. With this notion of morphism, the class of all associative algebras over *K* becomes a category.

Take for example the algebra *A* of all real-valued continuous functions **R** `->` **R**, and *B* = **R**. Both are algebras over **R**, and the map which assigns to every continuous function φ the number φ(0) is an algebra homomorphism from *A* to *B*.

One may consider associative algebras over a commutative ring *R*: these are modules over *R* together with a *R*-bilinear map which yields an associative multiplication.

The *n*-by-*n* matrices with integer entries form an associative algebra over the integers and the polynomials with coefficients in the ring **Z**_{n} (see modular arithmetic) form an associative algebra over **Z**_{n}.