Given a module 
over a commutative unit ring 
and a filtration
 |
(1)
|
of ideals of 
, the associated graded module of 
with respect to 
is
 |
(2)
|
which is a graded module over the associated graded ring 
with respect to the addition and the multiplication by scalars defined componentwise.
If 
is a proper ideal of 
, then the notation 
indicates the associated graded module of 
with respect to the 
-adic filtration of 
,
 |
(3)
|
