In mathematics, an

**associative algebra**is a vector space (or more generally module) which also allows the multiplication of vectors in a distributive and associative manner. They are thus special algebras.

Table of contents |

2 Examples 3 Algebra homomorphisms 4 Generalizations 5 Coalgebras |

### 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:

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

*A*contains an identity element, i.e. an element 1 such that 1

*x*=

*x*1 =

*x*for all

*x*in

*A*, then we call

*A*an

*associative algebra with one*or a

*unitary*(or

*unital*)

*associative algebra*. Such an algebra is a ring and contains a copy of the ground field

*K*in the form {

*a*1 :

*a*in

*K*}.

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

### Examples

- 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 (but not an algebra over the complex numbers, since complex numbers don't commute with quaternions).
- 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 (using composition of operators as multiplication); 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; here we add and multiply functions pointwise. - An example of a non-unitary associative algebra is given by the set of all functions
*f*:**R**`->`**R**whose limit as*x*nears infinity is zero. - The Clifford algebras are useful in geometry and physics.
- Incidence algebras of locally finite partially ordered sets are unitary associative algebras considered in combinatorics.

### Algebra homomorphisms

If *A* and *B* are associative algebras over the same field *K*, an *algebra homomorphism* *h*: *A* `->` *B* is a *K*-linear map which is also multiplicative in the sense that *h*(*xy*) = *h*(*x*) *h*(*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 *f* the number *f*(0) is an algebra homomorphism from *A* to *B*.

### Generalizations

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. In this case, a unitary *R*-algebra *A* can equivalently be defined as a ring *A* with a ring homomorphism *R*→*A*.

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***Z** (see modular arithmetic) form an associative algebra over **Z**/*n***Z**.

### Coalgebras

An associative unitary algebra over *K* is based on a morphism *A*×*A*→*A* having 2 inputs (multiplicator and multiplicand) and one output (product), as well as a morphism *K*→*A* identifying the scalar multiples of the multiplicative identity. These two morphisms can be dualized using categorial duality by reversing all arrows in the commutative diagrams which describe the algebra axioms; this defines the structure of a coalgebra.