The Rees module of a module 
over a commutative unit ring 
with respect to a filtration
 |
(1)
|
of ideals of 
is
 |
(2)
|
which is the set of finite formal polynomials in the variable 
in which the coefficient of
belongs to 
, the set of all finite sums of
terms 
with 
and 
. It is a graded module
over the Rees ring 
.
The subscript 
distinguishes it from the so-called extended Rees module, defined as
 |
(3)
|
where 
and hence 
for all 
.
Its elements are finite formal sums in integer
powers of 
whose coefficient of 
lies in 
.
If 
is a proper
ideal of 
,
the notation 
(or 
) indicates the (extended) Rees
module of 
with respect to the 
-adic
filtration.
