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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.

Table of contents
1 Definition
2 Examples
3 Algebra homomorphisms
4 Generalizations
5 Coalgebras


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: The bilinearity of the multiplication can be expressed as 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.


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.


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.


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.