A zero divisor is a nonzero element 
of a ring for
which 
or 
for some nonzero element 
, possibly 
itself. The first condition makes 
a left zero divisor and the second makes it a right zero divisor.
These conditions coincide in a commutative ring.
A commutative ring with multiplicative
identity 
and no zero divisors is an integral domain. Let
denote a nonzero 
-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 commutative ring with multiplicative
identity 
satisfying the zero product property.
