In abstract algebra, if I and J are ideals of a commutative ring R, their ideal quotient (I : J) is the set
(
I
:
J
)
=
{
r
∈
R
∣
r
J
⊂
I
}
Then (I : J) is itself an ideal in R. The ideal quotient is viewed as a quotient because
I
J
⊂
K
if and only if
I
⊂
K
:
J
. The ideal quotient is useful for calculating primary decompositions. It also arises in the description of the set difference in algebraic geometry (see below).
(I : J) is sometimes referred to as a colon ideal because of the notation. In the context of fractional ideals, there is a related notion of the inverse of a fractional ideal.
The ideal quotient satisfies the following properties:
(
I
:
J
)
=
A
n
n
R
(
(
J
+
I
)
/
I
)
as
R
-modules, where
A
n
n
R
(
M
)
denotes the annihilator of
M
as an
R
-module.
J
⊂
I
⇒
I
:
J
=
R
I
:
R
=
I
R
:
I
=
R
I
:
(
J
+
K
)
=
(
I
:
J
)
∩
(
I
:
K
)
I
:
(
r
)
=
1
r
(
I
∩
(
r
)
)
(as long as R is an integral domain)
The above properties can be used to calculate the quotient of ideals in a polynomial ring given their generators. For example, if I = (f1, f2, f3) and J = (g1, g2) are ideals in k[x1, ..., xn], then
I
:
J
=
(
I
:
(
g
1
)
)
∩
(
I
:
(
g
2
)
)
=
(
1
g
1
(
I
∩
(
g
1
)
)
)
∩
(
1
g
2
(
I
∩
(
g
2
)
)
)
Then elimination theory can be used to calculate the intersection of I with (g1) and (g2):
I
∩
(
g
1
)
=
t
I
+
(
1
−
t
)
(
g
1
)
∩
k
[
x
1
,
…
,
x
n
]
,
I
∩
(
g
2
)
=
t
I
+
(
1
−
t
)
(
g
1
)
∩
k
[
x
1
,
…
,
x
n
]
Calculate a Gröbner basis for tI + (1-t)(g1) with respect to lexicographic order. Then the basis functions which have no t in them generate
I
∩
(
g
1
)
.
The ideal quotient corresponds to set difference in algebraic geometry. More precisely,
If W is an affine variety and V is a subset of the affine space (not necessarily a variety), then
I
(
V
)
:
I
(
W
)
=
I
(
V
∖
W
)
where
I
(
∙
)
denotes the taking of the ideal associated to a subset.
If I and J are ideals in k[x1, ..., xn], with k algebraically closed and I radical then
Z
(
I
:
J
)
=
c
l
(
Z
(
I
)
∖
Z
(
J
)
)
where
c
l
(
∙
)
denotes the Zariski closure, and
Z
(
∙
)
denotes the taking of the variety defined by an ideal. If I is not radical, then the same property holds if we saturate the ideal J:
Z
(
I
:
J
∞
)
=
c
l
(
Z
(
I
)
∖
Z
(
J
)
)
where
(
I
:
J
∞
)
=
∪
n
≥
1
(
I
:
J
n
)
.
One geometric application of the ideal quotient is removing an irreducible component of an affine scheme. For example, let
I
=
(
x
y
z
)
,
J
=
(
x
y
)
in
C
[
x
,
y
,
z
]
be the ideals corresponding to the union of the x,y, and z-planes and x and y planes in
A
C
3
. Then, the ideal quotient
(
I
:
J
)
=
(
z
)
is the ideal of the z-plane in
A
C
3
. This shows how the ideal quotient can be used to "delete" irreducible subschemes.
A useful scheme theoretic example is taking the ideal quotient of a reducible ideal. For example, the ideal quotient
(
(
x
4
y
3
)
:
(
x
2
y
2
)
)
=
(
x
2
y
)
, showing that the ideal quotient of a subscheme of some non-reduced scheme, where both have the same reduced subscheme, kills off some of the non-reduced structure.
We can use the previous example to find the saturation of an ideal corresponding to a projective scheme. Given a homogeneous ideal
I
⊂
R
[
x
0
,
…
,
x
n
]
the saturation of
I
is defined as the ideal quotient
(
I
:
m
∞
)
=
∪
i
≥
1
(
I
:
m
i
)
where
m
=
(
x
0
,
…
,
x
n
)
⊂
R
[
x
0
,
…
,
x
n
]
. It is a theorem that the set of saturated ideals of
R
[
x
0
,
…
,
x
n
]
contained in
m
is in bijection with the set of projective subschemes in
P
R
n
. This shows us that
(
x
4
+
y
4
+
z
4
)
m
k
defines the same projective curve as
(
x
4
+
y
4
+
z
4
)
in
P
C
2
.
