In mathematics, the associated graded ring of a ring R with respect to a proper ideal I is the graded ring:
gr
I
R
=
⊕
n
=
0
∞
I
n
/
I
n
+
1
.
Similarly, if M is a left R-module, then the associated graded module is the graded module over
gr
I
R
:
gr
I
M
=
⊕
0
∞
I
n
M
/
I
n
+
1
M
.
Basic definitions and properties
For a ring R and ideal I, multiplication in
gr
I
R
is defined as follows: First, consider homogeneous elements
a
∈
I
i
/
I
i
+
1
and
b
∈
I
j
/
I
j
+
1
and suppose
a
′
∈
I
i
is a representative of a and
b
′
∈
I
j
is a representative of b. Then define
a
b
to be the equivalence class of
a
′
b
′
in
I
i
+
j
/
I
i
+
j
+
1
. Note that this is well-defined modulo
I
i
+
j
+
1
. Multiplication of inhomogeneous elements is defined by using the distributive property.
A ring or module may be related to its associated graded ring or module through the initial form map. Let M be an R-module and I an ideal of R. Given
f
∈
M
, the initial form of f in
gr
I
M
, written
i
n
(
f
)
, is the equivalence class of f in
I
m
M
/
I
m
+
1
M
where m is the maximum integer such that
f
∈
I
m
M
. If
f
∈
I
m
M
for every m, then set
i
n
(
f
)
=
0
. The initial form map is only a map of sets and generally not a homomorphism. For a submodule
N
⊂
M
,
i
n
(
N
)
is defined to be the submodule of
gr
I
M
generated by
{
i
n
(
f
)
|
f
∈
N
}
. This may not be the same as the submodule of
gr
I
M
generated by the only initial forms of the generators of N.
A ring inherits some "good" properties from its associated graded ring. For example, if R is a noetherian local ring, and
gr
I
R
is an integral domain, then R is itself an integral domain.
Let U be the enveloping algebra of a Lie algebra
g
over a field k; it is filtered by degree. The Poincaré–Birkhoff–Witt theorem implies that
gr
U
is a polynomial ring; in fact, it is the coordinate ring
k
[
g
∗
]
.
The associated graded algebra of a Clifford algebra is an exterior algebra; i.e., a Clifford algebra degenerates to an exterior algebra.
The associated graded can also be defined more generally for multiplicative descending filtrations of R (see also filtered ring.) Let F be a descending chain of ideals of the form
R
=
I
0
⊃
I
1
⊃
I
2
⊃
⋯
such that
I
j
I
k
⊂
I
j
+
k
. The graded ring associated with this filtration is
gr
F
R
=
⊕
n
=
0
∞
I
n
/
I
n
+
1
. Multiplication and the initial form map are defined as above.
