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# Associative algebra

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.

### 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.
The bilinearity of the multiplication can be expressed as
• (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.
If A contains an identity element, i.e. an element 1 such that 1x = x1 = 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 {a1 : 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 RA.

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

### Coalgebras

An associative unitary algebra over K is based on a morphism A×AA having 2 inputs (multiplicator and multiplicand) and one output (product), as well as a morphism KA 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.