A nonzero element 
of a ring for which 
, where 
is some other nonzero element
and the multiplication 
is the multiplication of the ring. A ring
with no zero divisors is known as an integral domain.
Let 
denote an 
-algebra, so that 
is a vector space over 
and
 |
(1)
|
 |
(2)
|
Now define
 |
(3)
|
where 
.
is said to be 
-associative if there exists
an 
-dimensional subspace 
of 
such that 
for all 
and 
. 
is said to be tame if 
is a finite union of subspaces
of 
.
The zero product property is intimately tethered to the notion of a zero divisor. For example, one may equivalently define an integral domain as a ring which satisfies the zero product property.
