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integral closure of an ideal ：ウィキペディア英語版

integral closure of an ideal
In algebra, the integral closure of an ideal ''I'' of a commutative ring ''R'', denoted by

\overline

, is the set of all elements ''r'' in ''R'' that are integral over ''I'': there exist

a_i \in I^i

such that
:

r^n + a_1 r^ + \cdots + a_ 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

\overline

if and only if there is a finitely generated ''R''-module ''M'', annihilated only by zero, such that

r M \subset I M

. It follows that

\overline

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 = \overline

.
The integral closure of an ideal appears in a theorem of Rees that characterizes an analytically unramified ring.
== Examples ==

*In

\mathbb(y)

x^i y^

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.
*Let

R = k(\ldots, 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^ \cdots X_n^

. The integral closure of a monomial ideal is monomial.

抄文引用元・出典: フリー百科事典『ウィキペディア（Wikipedia）』
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