6.2 Integrally closed domains and their properties
2 min readโขjuly 25, 2024
and integrally closed domains are key concepts in ring theory. They help us understand how elements from larger rings relate to smaller ones, particularly when dealing with polynomial equations and field extensions.
Integrally closed domains have special properties like "lying over" and "going up." These properties give us insights into how prime ideals behave in ring extensions, which is crucial for understanding the structure of more complex algebraic objects.
Integral Extensions and Integrally Closed Domains
Definition of integrally closed domains
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Integrally closed domains
Domain R integrally closed in K when elements of K integral over R already in R
R contains all roots of monic polynomials with coefficients in R (polynomial rings)
Integral extensions
Extension of rings A โ B where every element of B integral over A
integrally closed in its field of fractions
Integral elements
Element b โ B integral over A satisfies equation with coefficients in A
Assume x in field of fractions and integral over UFD
Express x as fraction of in UFD
Use monic polynomial equation satisfied by x
Show denominator must divide numerator
Conclude x in UFD
Fields
Every field integrally closed in itself
is the field itself
Fields trivially integrally closed
Properties and Characterizations of Integrally Closed Domains
Properties of integrally closed domains
For every P of R, prime ideal Q of exists such that P = Q โฉ R
Integrally closed domains satisfy lying over property
Given prime ideals P โ P' in R and prime ideal Q in integral closure of R with Q โฉ R = P, prime ideal Q' in integral closure exists such that Q โ Q' and Q' โฉ R = P'
Integrally closed domains satisfy going up property
Domain integrally closed if and only if it satisfies both lying over and going up properties
Integral closure in fraction fields
Integral closure
Set of all elements in field of fractions integral over domain
R denotes integral closure of R in its field of fractions
Properties of integral closure
R integrally closed
R โ R โ K, K field of fractions of R
R already integrally closed, then R = R
Constructing integral closure
Adjoin all integral elements from field of fractions to original domain
Iterate process if necessary (transfinite induction in some cases)
Applications
takes integral closure to obtain integrally closed domain
Studies singularities in algebraic geometry (algebraic curves)
Key Terms to Review (14)
Coprime Elements: Coprime elements, also known as relatively prime or mutually prime elements, are two elements in a ring such that their greatest common divisor (gcd) is one. This means that they share no common factors other than the unit elements of the ring. In the context of integrally closed domains, coprimeness relates to how ideals can be factored and how certain properties like unique factorization are preserved.
Equivalence: Equivalence refers to a relation that defines when two elements are considered equal in a specific context, often involving properties that are preserved under certain operations. In the context of integrally closed domains, equivalence can relate to how elements interact with each other in terms of being integral over a subring, leading to critical insights about the structure of these domains. Understanding equivalence helps in exploring concepts like integrality, closure properties, and factorization.
Field of Fractions: A field of fractions is a construction that takes an integral domain and creates a field where every element can be expressed as a fraction with a numerator and a denominator from the original integral domain, except for zero. This construction allows us to perform division by non-zero elements, facilitating the exploration of properties that are not always available in the original domain. The concept is key in understanding how integral domains relate to fields, especially when discussing prime and maximal ideals, localization, and properties of integrally closed domains.
Field of fractions of a field: The field of fractions of a field is a construction that allows us to form a new field by including all possible quotients of elements from the original field. This concept is essential for understanding the relationships between different fields, especially in terms of their integrally closed properties and how they relate to algebraic structures. It plays a crucial role in extending fields and analyzing their properties, particularly when dealing with integral elements and their corresponding domains.
Going Up Property: The going up property is a fundamental characteristic of integrally closed domains that asserts if a ring contains an element that is integral over a subring, then that element must be contained in the integral closure of that subring. This property is crucial for understanding the behavior of rings and their extensions, particularly in how they relate to integral elements and their containment within larger structures.
Integral closure of r: The integral closure of a ring r is the set of elements in its field of fractions that are integral over r. This means they satisfy a monic polynomial equation with coefficients from r. Integral closure helps in understanding the behavior of rings under extensions and is crucial for determining when a ring is integrally closed, which has significant implications for the properties of algebraic structures.
Integral Element: An integral element over a ring is an element in an extension ring that satisfies a polynomial equation with coefficients from the original ring. This concept is crucial for understanding the relationship between rings and their extensions, especially when examining properties like integral closure and integrally closed domains.
Integral Extensions: Integral extensions are a type of ring extension where each element of the extension ring is integral over the base ring. This means that for any element in the extension, there exists a monic polynomial with coefficients from the base ring such that the element is a root of that polynomial. Understanding integral extensions is crucial for exploring integrally closed domains, which are rings that coincide with their integral closures, and for studying properties related to factorization and local rings.
Integrally closed domain: An integrally closed domain is a type of integral domain where every element that is integral over the domain actually lies within the domain itself. This means that if an element satisfies a monic polynomial with coefficients in the domain, then that element must be part of the domain. This concept is closely linked to prime and maximal ideals, since being integrally closed can help determine properties of these ideals and how they interact with the structure of the ring.
Lying over property: The lying over property refers to a situation in ring theory where, given a ring homomorphism from a local ring to another local ring, any prime ideal in the second ring that lies over a prime ideal in the first ring contains a prime ideal that is the image of the first one. This concept is crucial when discussing how properties of ideals behave under ring extensions and is particularly relevant when understanding integral extensions and the behavior of prime ideals during these transitions.
Monic polynomial: A monic polynomial is a polynomial in which the leading coefficient (the coefficient of the highest degree term) is equal to 1. This property simplifies many aspects of polynomial algebra, especially when considering roots and factorization, making them particularly useful in various algebraic contexts, such as in integrally closed domains.
Normalization Process: The normalization process refers to the method of obtaining the integral closure of a ring, particularly in the context of integral domains. This process identifies elements that are integral over a given domain, ensuring that the resulting ring is integrally closed, meaning it contains all elements that are roots of monic polynomials with coefficients in that domain. This property is essential for studying various algebraic structures and their properties.
Prime Ideal: A prime ideal in a commutative ring is a proper ideal such that if the product of two elements is in the ideal, at least one of those elements must also be in the ideal. This concept is essential for understanding the structure of rings and has deep connections with other algebraic concepts, including maximal ideals and quotient rings. Prime ideals play a crucial role in defining prime elements and their relationship to irreducibility in algebraic structures.
Unique Factorization Domain: A unique factorization domain (UFD) is an integral domain in which every non-zero, non-unit element can be represented uniquely as a product of irreducible elements, up to order and units. This property makes UFDs similar to the integers, where numbers can be factored into prime numbers. In the context of ideals, integral domains, and integrally closed domains, unique factorization plays a key role in understanding the structure of these mathematical objects and how they relate to one another.
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