6.1 Prime ideals and maximal ideals
4 min read•july 30, 2024
Prime and maximal ideals are crucial in Dedekind domains. They're like the building blocks of these special rings, helping us understand their structure and properties. Think of them as the prime numbers of ideals, allowing us to factor and analyze more complex structures.
In Dedekind domains, are always maximal. This unique feature simplifies things, making it easier to work with ideals and study their relationships. It's a key tool for tackling problems in algebraic number theory and related fields.
Prime and Maximal Ideals in Dedekind Domains
Definitions and Key Concepts
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- defined as with three properties
- Noetherian
- Integrally closed
- Every non-zero maximal
- Prime ideal P in Dedekind domain R characterized by
- For , if , then or
- M in Dedekind domain R identified as
- Proper ideal
- Not contained in any other proper ideal of R
- Zero ideal prime but not maximal in Dedekind domains
- properties
- integral domain when P prime
- field when M maximal
- Prime ideals analogous to prime numbers in integer factorization
- Maximal ideals correspond to prime elements in
Examples and Applications
- Ring of integers exemplifies Dedekind domain
- Prime ideals form where p prime number (2, 3, 5, 7, 11, etc.)
- demonstrate complex Dedekind domain
- Prime ideals include and
- in number fields illustrate more complex cases
- contains non-principal prime ideals
- Polynomial rings over field K serve as Dedekind domains
- Prime ideals form where irreducible polynomial
- Coordinate rings of nonsingular algebraic curves over algebraically closed fields provide geometric examples
- Discrete valuation rings represent local Dedekind domains
- Principal ideal domains (PIDs) form special case of Dedekind domains
Relationship Between Prime and Maximal Ideals
Equivalence in Dedekind Domains
- Non-zero prime ideals always maximal in Dedekind domains
- Converse true maximal ideals always prime
- Equivalence defining characteristic of Dedekind domains
- Zero ideal sole exception to prime-maximal equivalence
- Simplifies ideal structure in Dedekind domains
- Enables of ideals into prime (or maximal) ideal products
- Ensures quotient ring by non-zero prime ideal always field
Implications and Applications
- Strengthens connection between ring theory and field theory
- Facilitates study of algebraic number fields
- Allows generalization of unique factorization from elements to ideals
- Simplifies proofs and computations in algebraic number theory
- Provides framework for studying in number field extensions
- Enables development of
- Connects to geometric interpretations in algebraic geometry
Properties of Prime and Maximal Ideals
Theoretical Foundations
- Non-zero prime ideals maximal in Dedekind domains
- Proof utilizes Dedekind domain definition and prime ideal properties
- Intersection of distinct maximal ideals equals their product
- Demonstrates ideal arithmetic in Dedekind domains
- Unique factorization of ideals as prime (or maximal) ideal products
- Generalizes unique factorization of integers
- characterized by their sum
- Two ideals coprime if and only if sum entire ring
- form group under multiplication
- Extends notion of ideal arithmetic
- Ideals generated by at most two elements
- Simplifies ideal representation
- at prime ideal yields
- Connects global and local properties
Practical Applications
- Simplifies computations in algebraic number theory
- Factoring ideals analogous to factoring integers
- Enables study of ramification in number field extensions
- Prime ideal factorization reveals splitting behavior
- Facilitates development of ideal class group theory
- Measures how far ring deviates from being PID
- Provides tools for solving Diophantine equations
- Ideal factorization aids in finding integer solutions
- Connects to algebraic geometry through schemes
- Prime ideals correspond to points in geometric space
- Applies to cryptography and coding theory
- Ideal structure used in certain encryption schemes
Identifying Ideals in Dedekind Domains
Techniques and Strategies
- Analyze factorization of elements to identify prime ideals
- In , prime ideals form where p prime (2, 3, 5, 7, etc.)
- Examine norm of elements in algebraic number fields
- Helps identify prime ideals in rings of integers
- Use irreducibility tests for polynomials in
- Prime ideals generated by irreducible polynomials
- Apply to study ideal structure
- Aids in understanding relationships between ideals
- Utilize for ramification analysis
- Determines prime ideal factorization in extensions
- Employ localization techniques to study ideal behavior
- Reduces global problems to local ones
- Leverage for certain number fields
- Connects ideal structure to field automorphisms
Common Examples and Special Cases
- Ring of integers serves as simplest Dedekind domain
- All ideals principal, generated by single element
- Gaussian integers illustrate complex case
- Prime ideals include and
- Algebraic integers in demonstrate non-UFD case
- Contains non-principal ideals
- over finite field exemplifies positive characteristic
- Prime ideals generated by irreducible polynomials
- Coordinate ring of elliptic curve provides geometric example
- Prime ideals correspond to points on curve
- Discrete valuation rings represent local Dedekind domains
- Unique maximal ideal, all ideals powers of maximal ideal
- PIDs form special case where all ideals principal
- Simplifies ideal structure and computations
Key Terms to Review (27)
Algebraic Integers: Algebraic integers are complex numbers that are roots of monic polynomials with integer coefficients. They play a crucial role in number theory, particularly in the study of unique factorization and properties of number fields, connecting various concepts like integral bases and prime ideals.
Chinese Remainder Theorem: The Chinese Remainder Theorem is a fundamental result in number theory that provides a way to solve systems of simultaneous congruences with pairwise coprime moduli. This theorem states that if you have multiple congruences, you can find a unique solution modulo the product of these moduli. It connects to prime ideals and maximal ideals by showing how the structure of integers can be analyzed through these ideals, and it relates to the local-global principle by illustrating how local solutions can lead to a global solution.
Coprimality of ideals: Coprimality of ideals refers to the situation where two ideals in a ring do not share any common factors, meaning that their intersection is as small as possible. This concept plays a significant role in understanding the structure of rings and their ideals, especially in distinguishing between prime and maximal ideals. When ideals are coprime, it can lead to useful decompositions and simplifications in algebraic problems.
Correspondence theorem: The correspondence theorem is a fundamental result in ring theory that establishes a relationship between ideals in a ring and ideals in its quotient ring. It helps to clarify how prime ideals and maximal ideals correspond under the process of taking quotients, highlighting how these structures are preserved or altered when moving from a ring to its quotient.
David Hilbert: David Hilbert was a prominent German mathematician known for his foundational contributions to various fields, including algebra, number theory, and mathematical logic. His work laid the groundwork for modern mathematics and significantly influenced the development of algebraic number theory.
Dedekind domain: A Dedekind domain is a type of integral domain in which every non-zero proper ideal can be uniquely factored into a product of prime ideals. This property allows Dedekind domains to generalize many familiar concepts in number theory, such as the ring of integers and unique factorization, while also providing a framework for understanding fractional ideals and ideal class groups.
Dedekind-Kummer Theorem: The Dedekind-Kummer Theorem is a fundamental result in algebraic number theory that relates the behavior of prime ideals in a number field to the roots of unity and the properties of the field's discriminant. This theorem establishes a connection between the discriminant of a number field and the ramification of primes in its extension fields, helping to understand how primes factor in these extensions.
Discrete Valuation Ring: A discrete valuation ring (DVR) is a principal ideal domain that is also a local ring with a unique non-zero maximal ideal. This structure allows for a well-defined valuation, which assigns to each non-zero element of the ring a unique integer that describes its divisibility properties. The DVR is significant because it provides a framework for understanding how prime ideals behave within local contexts, particularly in algebraic number theory.
Emil Artin: Emil Artin was a prominent 20th-century mathematician known for his significant contributions to algebraic number theory, particularly in the areas of class field theory and algebraic integers. His work has influenced various aspects of modern mathematics, linking concepts like field extensions and ideals to the broader framework of number theory.
Fractional Ideals: Fractional ideals are a generalization of ideals in the context of algebraic number theory, allowing for the representation of elements in a number field that may not be integral. They provide a framework to analyze divisibility and factorization properties within the ring of integers of a number field, linking them closely to the concepts of prime and maximal ideals as well as class groups.
Galois Theory: Galois Theory is a branch of mathematics that connects field theory and group theory, providing a framework to understand the symmetries of the roots of polynomial equations. It explores how the structure of field extensions relates to the properties of groups, especially focusing on the relationships between subfields and subgroups. This theory serves as a fundamental tool for determining when a polynomial can be solved by radicals and plays a crucial role in understanding the solvability of polynomial equations.
Gaussian Integers: Gaussian integers are complex numbers of the form $$a + bi$$ where both $$a$$ and $$b$$ are integers, and $$i$$ is the imaginary unit satisfying $$i^2 = -1$$. They form a unique ring that extends the concept of integers to include imaginary units, allowing for a rich structure where concepts like factorization and primality can be studied similarly to traditional integers.
Ideal Class Group Theory: Ideal class group theory is a framework in algebraic number theory that examines the properties of ideals in a number field and their role in determining the arithmetic structure of the ring of integers within that field. This theory primarily focuses on the classification of fractional ideals, allowing mathematicians to understand how these ideals relate to each other, especially through equivalence classes. In particular, it provides a way to measure the failure of unique factorization in the ring of integers, with the ideal class group itself serving as a crucial tool for understanding prime and maximal ideals.
Ideal Factorization: Ideal factorization refers to the process of expressing an ideal in a ring as a product of prime ideals, similar to how integers can be expressed as a product of prime numbers. This concept is crucial for understanding the structure of rings of integers and algebraic integers, where it reveals how ideals behave in relation to one another and how they can be decomposed within larger number fields or rings.
Integral Domain: An integral domain is a type of commutative ring with no zero divisors and a multiplicative identity, where the cancellation law holds. This means that in an integral domain, if the product of two elements is zero, at least one of those elements must be zero. Integral domains are essential for studying unique factorization and prime elements, as they provide a structured environment for exploring these concepts.
Intersection of Ideals: The intersection of ideals in a ring is the set of elements that are common to two or more ideals within that ring. This intersection itself forms an ideal, which allows for the exploration of relationships between ideals, including how they can influence properties such as prime and maximal ideals, and their respective structures within the context of ring theory.
Localization: Localization is a mathematical technique used to study properties of rings and modules by focusing on a particular subset of their elements, often in the context of prime or maximal ideals. It allows us to 'zoom in' on behavior around these ideals, enabling analysis of fractions and local properties that can simplify problems in algebraic structures. This is especially important when working with prime ideals, which help identify where certain divisibility properties hold true, and maximal ideals, which relate closely to field extensions and unique factorization.
Maximal Ideal: A maximal ideal is an ideal in a ring that is proper and maximal with respect to inclusion, meaning it cannot be contained in any larger proper ideal. This concept connects to important structures in algebraic systems, helping in the classification of rings and understanding algebraic integers and numbers, as well as paving the way to comprehend prime ideals and their unique properties within a ring.
Non-zero prime ideals: Non-zero prime ideals are a special type of ideal in a ring that are not only prime but also do not contain the zero element. These ideals play a crucial role in understanding the structure of rings, as they help identify the fundamental building blocks within the algebraic framework. The properties of non-zero prime ideals are closely tied to the notions of irreducibility and factorizations in the context of rings.
Polynomial Ring: A polynomial ring is a mathematical structure formed by polynomials with coefficients from a given ring, which allows for addition and multiplication of these polynomials. This structure is crucial for understanding the behavior of polynomials in various algebraic contexts, especially regarding ideals, as it provides a natural setting to discuss concepts like prime and maximal ideals.
Prime Ideal: A prime ideal in a ring is a special type of ideal such that if the product of two elements is in the ideal, then at least one of those elements must also be in the ideal. This concept helps to identify the structure and properties of rings, especially in understanding the relationships between ideals and factors in rings, connecting closely with notions like unique factorization.
Principal Ideal Domain: A principal ideal domain (PID) is an integral domain in which every ideal is principal, meaning it can be generated by a single element. This property makes PIDs a crucial concept in ring theory, particularly for understanding the structure of ideals and their relationship to prime and maximal ideals. In a PID, not only do unique factorization properties often hold, but the relationships between prime ideals and maximal ideals can be clearly delineated.
Proper Ideal: A proper ideal is a subset of a ring that is itself an ideal and is not equal to the entire ring. Proper ideals play a crucial role in understanding the structure of rings, particularly in the context of prime and maximal ideals, as they help define the relationships between different types of ideals and their properties in ring theory.
Quotient Ring: A quotient ring is a type of algebraic structure formed from a ring by partitioning it into equivalence classes using a congruence relation defined by an ideal. In simpler terms, when you take a ring and 'mod out' by one of its ideals, the result is a quotient ring, which captures the idea of working with the original ring while ignoring certain elements related to the ideal. This concept is crucial for understanding properties such as prime ideals and maximal ideals, as these types of ideals influence the structure and behavior of the resulting quotient rings.
Ramification: Ramification refers to how primes in a base field split or remain inert when extended to a larger field. It highlights the behavior of prime ideals under field extensions, particularly focusing on their splitting, degree of extension, and how they relate to the discriminant. This concept is crucial for understanding the structure of number fields and how they behave under various algebraic operations.
Unique Factorization: Unique factorization refers to the property of integers and certain algebraic structures where every element can be expressed uniquely as a product of irreducible elements, up to ordering and units. This concept is crucial in understanding the structure of rings and fields, as it establishes a foundational aspect of number theory that extends into the realm of algebraic number theory, where unique factorization might not hold in every context.
Zorn's Lemma: Zorn's Lemma is a proposition in set theory that states if a partially ordered set has the property that every chain (totally ordered subset) has an upper bound, then the entire set contains at least one maximal element. This concept is fundamental in various areas of mathematics, particularly in proving the existence of algebraic closures and understanding the structure of prime ideals in ring theory.