In algebra, the integral closure of an ideal I of a commutative ring R, denoted by
I
¯
, is the set of all elements r in R that are integral over I: there exist
a
i
∈
I
i
such that
r
n
+
a
1
r
n
−
1
+
⋯
+
a
n
−
1
r
+
a
n
=
0.
It is similar to the integral closure of a subring. For example, if R is a domain, an element r in R belongs to
I
¯
if and only if there is a finitely generated R-module M, annihilated only by zero, such that
r
M
⊂
I
M
. It follows that
I
¯
is an ideal of R (in fact, the integral closure of an ideal is always an ideal; see below.) I is said to be integrally closed if
I
=
I
¯
.
The integral closure of an ideal appears in a theorem of Rees that characterizes an analytically unramified ring.
In
C
[
x
,
y
]
,
x
i
y
d
−
i
is integral over
(
x
d
,
y
d
)
.
Radical ideals (e.g., prime ideals) are integrally closed. The intersection of integrally closed ideals is integrally closed.
In a normal ring, for any non-zerodivisor x and any ideal I,
x
I
¯
=
x
I
¯
. In particular, in a normal ring, a principal ideal generated by a non-zerodivisor is integrally closed.
Let
R
=
k
[
X
1
,
…
,
X
n
]
be a polynomial ring over a field k. An ideal I in R is called monomial if it is generated by monomials; i.e.,
X
1
a
1
⋯
X
n
a
n
. The integral closure of a monomial ideal is monomial.
Let R be a ring. The Rees algebra
R
[
I
t
]
=
⊕
n
≥
0
I
n
t
n
can be used to compute the integral closure of an ideal. The structure result is the following: the integral closure of
R
[
I
t
]
in
R
[
t
]
, which is graded, is
⊕
n
≥
0
I
n
¯
t
n
. In particular,
I
¯
is an ideal and
I
¯
=
I
¯
¯
; i.e., the integral closure of an ideal is integrally closed. It also follows that the integral closure of a homogeneous ideal is homogeneous.
The following type of results is called the Briancon–Skoda theorem: let R be a regular ring and I an ideal generated by l elements. Then
I
n
+
l
¯
⊂
I
n
+
1
for any
n
≥
0
.
A theorem of Rees states: let (R, m) be a noetherian local ring. Assume it is formally equidimensional (i.e., the completion is equidimensional.). Then two m-primary ideals
I
⊂
J
have the same integral closure if and only if they have the same multiplicity.