1 Basic definitions

Integral closure is built from the notion of an element satisfying a monic polynomial over a smaller ring. It provides a way to compare a ring with a larger ambient ring and identify which elements behave algebraically over the original one.

1.1 Integral elements

Let \(A \subseteq B\) be rings, and let \(b \in B\). The element \(b\) is called integral over \(A\) if there exists a monic polynomial \[ b^n + a_{n-1}b^{n-1} + \cdots + a_1 b + a_0 = 0 \] with all coefficients \(a_i \in A\). Such an element is said to satisfy an integral dependence relation over \(A\).

Integral elements often arise naturally in algebraic settings, especially when a larger ring is generated by algebraic data over a smaller one. The condition is stronger than mere algebraicity over a field, because the coefficients must lie in the ring itself.

1.2 Integral extensions

A ring extension \(A \subseteq B\) is called integral if every element of \(B\) is integral over \(A\). This property indicates that \(B\) is built from \(A\) using only algebraic elements of a controlled type.

Integral extensions preserve many finiteness features. In particular, they often allow one to reduce questions about \(B\) to questions about \(A\), especially when \(B\) is finitely generated as an \(A\)-module.

1.3 Integral closure of a subring

Given a subring \(A \subseteq B\), the integral closure of \(A\) in \(B\) is the set of all elements of \(B\) that are integral over \(A\). It is itself a subring of \(B\), and it is the largest subring of \(B\) that is integral over \(A\).

This construction measures how much of \(B\) can be recovered from \(A\) by adjoining algebraic elements that satisfy monic equations with coefficients in \(A\). In many settings, the integral closure is the natural “completed” version of \(A\) inside \(B\).

1.4 Integral closure of an integral domain

If \(A\) is an integral domain with field of fractions \(K\), then the integral closure of \(A\) in \(K\) consists of all elements of \(K\) integral over \(A\). This ring is often compared with \(A\) to determine whether \(A\) is already closed under integral dependence.

When \(A\) equals its own integral closure in \(K\), the domain is called integrally closed. This is a key regularity condition in commutative algebra and algebraic geometry.

2 Fundamental properties

Integral closure has several stability properties that make it useful in structural arguments. These include closure under arithmetic operations, compatibility with localization, and good behavior under composition of extensions.

2.1 Closure under ring operations

The set of elements integral over a given ring is closed under addition and multiplication, so it forms a subring. It also contains the image of the base ring and is stable under taking powers.

This closure property is essential: it ensures that integral elements do not appear in isolation but assemble into a meaningful ring-theoretic object. It also allows one to form the integral closure without leaving the category of rings.

2.2 Transitivity

If \(A \subseteq B \subseteq C\), and \(B\) is integral over \(A\) while \(C\) is integral over \(B\), then \(C\) is integral over \(A\). This is the transitivity of integrality.

The property implies that integral dependence can be checked in stages. It is particularly useful when one adjoins elements one at a time, since each step preserves integrality relative to the original ring.

2.3 Behavior under localization

Integral closure behaves well under localization. If \(S\) is a multiplicative set in \(A\), then localizing an integral extension \(A \subseteq B\) yields an integral extension \(S^{-1}A \subseteq S^{-1}B\).

This makes integral closure compatible with local analysis. Since many algebraic questions are local in nature, one can often study integral dependence prime by prime or near specific points of a scheme.

2.4 Behavior under quotient rings

Integrality is also compatible with quotient constructions in many standard situations. If \(b\) is integral over \(A\), then its image in a quotient ring is integral over the corresponding quotient of \(A\), provided the quotient is taken by a compatible ideal.

This property helps transfer integral dependence through algebraic reductions. It is frequently used when analyzing the structure of rings modulo ideals, especially in geometric and arithmetic applications.

3 Characterizations

Several equivalent criteria describe integral dependence. These characterizations are valuable because they connect polynomial equations, module-theoretic finiteness, and valuation-theoretic tests.

3.1 Monic polynomial criterion

The defining criterion for integrality is the existence of a monic polynomial relation with coefficients in the base ring. If an element satisfies such a relation, it is integral over that ring.

This criterion is direct and easy to apply in explicit examples. It is the starting point for most proofs involving integral closure.

3.2 Module-finiteness criterion

An element \(b\) is integral over \(A\) if and only if the subring \(A[b]\) is a finitely generated \(A\)-module. More generally, a ring extension \(A \subseteq B\) is integral if and only if each element of \(B\) generates a finite \(A\)-module upon adjoining.

This characterization is especially useful because it translates a polynomial condition into a finiteness statement. It links integral closure with the theory of finitely generated modules and Noetherian rings.

3.3 Valuative criteria

For domains, integrality can often be tested using valuation rings. An element of the fraction field is integral over a domain if it lies in every valuation ring containing the domain in the relevant fraction field.

Valuative criteria are central in algebraic geometry and birational methods. They provide a way to detect whether an element has poles or other singular behavior relative to the original ring.

4 Examples

Concrete examples show how integral closure operates in arithmetic and polynomial settings. They also illustrate the difference between rings that are already integrally closed and those that are not.

4.1 Integers in number fields

The ring of algebraic integers in a number field is the integral closure of \(\mathbb{Z}\) in that field. For example, in quadratic fields, one obtains rings such as \(\mathbb{Z}[\sqrt{d}]\) or closely related variants depending on the discriminant.

This example is foundational in algebraic number theory. It shows how integral closure generalizes ordinary integers to more complicated algebraic settings.

4.2 Polynomial rings and their extensions

Polynomial rings over a field are typically integrally closed, and many natural algebraic extensions can be analyzed by adjoining roots of monic polynomials. If a root is integral over the base ring, then the extension often reflects a geometric covering or a finite algebraic map.

Such examples include adjoining square roots, cube roots, or more general algebraic functions. They demonstrate how integral closure captures algebraic relationships without introducing denominators.

4.3 Integrally closed domains

A domain is integrally closed if it contains every element of its fraction field that is integral over it. Familiar examples include principal ideal domains, unique factorization domains, and polynomial rings over fields.

These rings are often structurally well behaved. Integrally closedness is an important hypothesis in many theorems, since it rules out certain hidden algebraic defects.

4.4 Non-integrally closed rings

A ring may fail to be integrally closed if it omits elements that satisfy monic relations over it. For instance, subrings of the form \(k[t^2,t^3]\) inside \(k[t]\) are not integrally closed, since \(t\) is integral over the subring but does not belong to it.

Such examples are common in singularity theory and in the study of affine curves. They show how integral closure can enlarge a ring in a controlled but nontrivial way.

5 Normalization

Normalization is the process of passing from a domain to its integral closure in the field of fractions. It is a standard tool for repairing algebraic defects and obtaining a more regular ambient ring.

5.1 Integral closure in the field of fractions

For an integral domain \(A\) with fraction field \(K\), the normalization of \(A\) is its integral closure in \(K\). This construction produces the smallest integrally closed domain containing \(A\) inside \(K\).

Normalization is often finite in important cases, especially for rings arising in algebraic geometry and number theory. It is the algebraic analogue of resolving certain kinds of incompleteness.

5.2 Normal rings and normal domains

A domain is normal if it is integrally closed in its fraction field. More generally, a reduced ring is often called normal if all of its localizations at prime ideals are integrally closed domains.

Normality is a central regularity condition. It is weaker than smoothness but strong enough to control many algebraic and geometric phenomena.

5.3 Relation to singularities

Normalization often improves singular behavior by separating and correcting certain algebraic pathologies. While it does not necessarily smooth a space completely, it can remove defects caused by missing integral elements.

In geometric terms, a nonnormal variety may acquire a better-behaved coordinate ring after normalization. This makes the process an important first step in the study of singular points.

6 Computation and construction

Integral closure can sometimes be computed explicitly, though the process may be technically demanding. Construction methods include adjoining integral elements and using finite module techniques.

6.1 Adjoining integral elements

One way to build the integral closure is to adjoin to the base ring all elements known to be integral over it. In practice, one often proceeds step by step, adding generators that satisfy monic equations.

This method is conceptually simple and works well in small examples. It also reflects the idea that normalization is obtained by systematically filling in missing integral points.

6.2 Finite integral extensions

If an extension is integral and finitely generated as a module, then its structure can often be described by generators and relations. Such extensions are especially manageable in computational settings.

Finite integral extensions are common in explicit algebraic constructions. They provide a controlled framework in which one can compute normalizations or test whether specific elements are integral.

6.3 Algorithms for integral closure

Computer algebra systems implement algorithms for integral closure in many classes of rings, especially affine algebras over fields or integers. These algorithms may use Gröbner bases, discriminants, trace forms, or local computations.

Although the general problem can be difficult, modern methods are effective in a wide range of examples. They are widely used in symbolic computation and computational algebraic geometry.

6.4 Examples of explicit normalization

For rings like \(k[t^2,t^3]\), normalization is obtained by adjoining \(t\), yielding \(k[t]\). More complicated plane curve singularities can be normalized by adjoining rational functions that remove cusps or nodes in the algebraic description.

These examples show the practical meaning of normalization. The resulting ring often has a simpler presentation and better structural properties.

7 Applications

Integral closure appears in several major areas of mathematics. It provides a common language for algebraic dependence, finite extensions, and the correction of singular behavior.

7.1 Algebraic geometry

In algebraic geometry, integral closure describes the normalization of affine varieties and schemes. It helps analyze morphisms that are finite and birational, and it plays a role in comparing a variety with its normalized version.

The concept is especially useful when studying coordinate rings of curves and higher-dimensional varieties. It links geometric regularity to algebraic completeness.

7.2 Algebraic number theory

In number theory, integral closure identifies rings of algebraic integers in finite extensions of \(\mathbb{Q}\). These rings serve as the natural domains for factorization and ideal theory in number fields.

Many arithmetic invariants are defined using the integral closure of \(\mathbb{Z}\) in such fields. This makes the concept fundamental to the structure of algebraic integers.

7.3 Study of singularities

Integral closure helps distinguish singular rings from normal ones. Since nonnormality can reflect hidden algebraic defects, normalization is an important step in analyzing and classifying singularities.

The process is often used in the local study of curves, surfaces, and higher-dimensional varieties. It provides a bridge between abstract algebra and geometric intuition.

7.4 Dimension and depth arguments

Integral extensions preserve many dimension-theoretic properties and can be used in arguments involving depth, height, and prime spectra. They often allow one to compare chains of prime ideals between a ring and its integral closure.

These comparisons are useful in commutative algebra, especially in Noetherian settings. They support structural results about local rings and their completions.

Integral closure is connected to several closely related notions in algebra and geometry. These ideas often appear together in the study of ring structure and geometric normalization.

8.1 Integrally closed subrings

An integrally closed subring is one that equals its own integral closure in a given ambient ring or fraction field. This property indicates that no additional integral elements are missing.

Such subrings are often used as reference points in comparative studies. They provide a natural notion of completeness with respect to integral dependence.

8.2 Dedekind domains

A Dedekind domain is a special kind of Noetherian, integrally closed domain of dimension one with additional factorization properties. These rings are central in algebraic number theory.

Their integrally closed nature makes ideal factorization especially well behaved. As a result, Dedekind domains often serve as a model case for arithmetic applications of integral closure.

8.3 Normalization map

The normalization map is the natural morphism from a ring or scheme to its integral closure or normalized space. In geometry, it typically induces a finite birational map from the normalization to the original object.

This map is a standard tool for comparing singular and improved versions of an algebraic structure. It encodes how integral closure modifies the underlying space.

8.4 Reduced and reduced closure notions

Reduced rings and related closure concepts help distinguish the removal of nilpotent elements from the addition of integral ones. While reducedness concerns the absence of nilpotents, integral closure concerns completeness under monic relations.

These notions are frequently studied together because both affect the geometry and arithmetic of a ring. Their interplay helps clarify which algebraic imperfections are present and how they may be corrected.