The associated graded ring of a commutative unit ring 
with respect to a filtration
 |
(1)
|
of ideals of 
is the graded ring
 |
(2)
|
The addition is defined componentwise, and the product is defined as follows. If ![a=[alpha]_i in I_i/I_(i+1)](/images/equations/AssociatedGradedRing/Inline3.svg)
is the residue
class of 
mod 
,
and ![b=[beta]_j in I_j/I_(j+1)](/images/equations/AssociatedGradedRing/Inline6.svg)
is the residue class of 
mod 
, then ![a·b=[alphabeta]_(i+j)](/images/equations/AssociatedGradedRing/Inline9.svg)
is the residue
class of 
mod 
.
is a quotient ring of the Rees
ring of 
with respect to 
,
 |
(3)
|
If 
is a proper ideal of 
, then the notation 
indicates the associated graded ring of 
with respect to the 
-adic filtration of 
,
 |
(4)
|
If 
is Noetherian, then 
is also Noetherian.
It is generated as a ring over its degree-zero copy of 
by finitely many classes in 
. If 
is a Noetherian ring that
is also a local ring with maximal
ideal 
,
and 
is its 
-adic
filtration, then
 |
(5)
|
