Integral closure of an ideal

In algebra, the integral closure of an ideal $I$ of a commutative ring $R$, denoted by $\bar{I}$, is the set of all elements r in $R$ that are integral over $I$: that is, for each $i$ there exists $a_i\in I^i$ such that

$r^n+a_1{r}^{n-1}+\cdots +a_{n-1}r+a_n=0.$

In other words, $r$ is a zero of a certain kind of monic polynomial. This integral closure is similar to the integral closure of a subring. For example, if $R$ is a domain, an element r in $R$ belongs to $\bar{I}$ if and only if there is a finitely generated $R$-module $M$, annihilated only by zero, such that $rM\subseteq IM$. It follows that $\bar{I}$ 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 $I=\bar{I}$.

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

Examples

• In $ℂ[x,y]$, $x^i{y}^{d-i}$ is integral over $(x^d,y^d)$. It satisfies the equation $r^d+(-x^{di}{y}^{d(d-i)})=0$, where $a_d=-x^{di}{y}^{d(d-i)}$ is in the $d$th power of the ideal.
• 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 $I$, $\overline{xI}=x\bar{I}$. In particular, in a normal ring, a principal ideal generated by a non-zerodivisor is integrally closed.
• Let $R=k[X_1,\dots ,X_n]$ be a polynomial ring over a field k. An ideal $I$ in $R$ is called monomial if it is generated by monomials; i.e., $X_1^{a_1}\cdots X_n^{a_n}$. The integral closure of a monomial ideal is monomial.

Structure results

Let $R$ be a ring. The Rees algebra $R[It]={\oplus }_{n\ge 0}{I}^n{t}^n$ can be used to compute the integral closure of an ideal. The structure result is the following: the integral closure of $R[It]$ in $R[t]$, which is graded, is ${\oplus }_{n\ge 0}\overline{I^n}{t}^n$. In particular, $\bar{I}$ is an ideal and $\bar{I}=\overline{\stackrel{‾}{I}}$; 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 Template:Mvar an ideal generated by Template:Mvar elements. Then $\overline{I^{n+l}}\subset I^{n+1}$ for any $n\ge 0$.

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 $I\subset J$ have the same integral closure if and only if they have the same multiplicity.

See also

• Dedekind–Kummer theorem

References

• Eisenbud, David, Commutative Algebra with a View Toward Algebraic Geometry, Graduate Texts in Mathematics, 150, Springer-Verlag, 1995, Template:ISBN.
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Further reading

• Irena Swanson, Rees valuations Template:Webarchive.