In simple terms, let 
, 
, and 
be members of an algebra. Then
the algebra is said to be associative if
 |
(1)
|
where 
denotes multiplication. More formally, let 
denote an 
-algebra, so that 
is a vector space over 
and
 |
(2)
|
 |
(3)
|
Then 
is said to be 
-associative
if there exists an 
-dimensional
subspace 
of 
such that
 |
(4)
|
for all 
and 
.
Here, vector multiplication 
is assumed to be bilinear.
An 
-dimensional
-associative
algebra is simply said to be "associative."
