A matrix factorization of a nonzero element 
, in square-matrix form, is a pair
of 
matrices 
and 
over a commutative ring 
with multiplicative
identity 
satisfying
 |
(1)
|
where 
is the identity matrix and 
is a positive integer.
This construction includes ordinary factorization
as the case 
and is useful even when 
is an irreducible polynomial
(Eisenbud and Peeva 2016, Numberphile 2020).
For example, over the polynomial ring ![R[x,y]](/images/equations/MatrixFactorization/Inline11.svg)
, the polynomial 
is irreducible,
but it has a matrix factorization with 
because
 |
(2)
|
Suppose in addition that 
is not a unit and is not a zero
divisor. Reducing the entries of 
and 
in the quotient ring 
gives matrices 
and 
whose products in both orders are zero. If 
is the cokernel of 
, then the alternating module
homomorphisms form a free resolution of the module 
over 
,
 |
(3)
|
The free module and module homomorphisms repeat with period at most two in positive degrees. This resolution need not be minimal.
To show that this is an exact sequence at a term followed by 
,
choose a lift 
of an element in its kernel.
Then 
for some 
,
so 
.
The matrix 
defines an injection on 
, since 
implies 
and 
is nonzero and not a zero
divisor. Thus 
,
proving that the kernel of 
equals the image of 
. The same argument with 
and 
interchanged applies at the other terms.
