Associative algebra

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

Defintion

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 with the properties

• (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.

Examples