In algebra, the integral closure of an ideal ''I'' of a commutative ring ''R'', denoted by
, is the set of all elements ''r'' in ''R'' that are integral over ''I'': there exist
such that
:
It is similar to the
integral closure In commutative algebra, an element ''b'' of a commutative ring ''B'' is said to be integral over ''A'', a subring of ''B'', if there are ''n'' ≥ 1 and ''a'j'' in ''A'' such that
:b^n + a_ b^ + \cdots + a_1 b + a_0 = 0.
That is to say, ''b'' ...
of a subring. For example, if ''R'' is a domain, an element ''r'' in ''R'' belongs to
if and only if there is a finitely generated ''R''-module ''M'', annihilated only by zero, such that
. It follows that
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
.
The integral closure of an ideal appears in a theorem of
Rees that characterizes an
analytically unramified ring.
Examples
*In