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 1
x =
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