An associative algebra is a vector space which also allows the multiplication of vectors in a distributive and associative manner.
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,
- α (x y) = (α x) y = x (α y) for all x, y in A and α in K.
If A contains an identity element, i.e. an element 1 such that 1x = x1 = 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.
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.
The n-by-n matrices with integer entries form an associative algebra over the integers and the polynomials with coefficients in the ring Zn (see modular arithmetic) form an associative algebra over Zn.