Dedekind Domain

Definition

A Dedekind domain is a type of integral domain where every non-zero prime ideal is maximal, and it satisfies the property that every fractional ideal can be expressed as a product of prime ideals. These domains are important in algebraic number theory and have connections to concepts like free and projective modules, integral elements and extensions, and regular sequences. Dedekind domains provide a framework for understanding unique factorization and the behavior of ideals in a way that is crucial for studying the arithmetic of algebraic integers.

5 Must Know Facts For Your Next Test

1. In a Dedekind domain, every non-zero proper ideal can be factored uniquely into a product of prime ideals, which extends the concept of unique factorization beyond integers.
2. Dedekind domains are always Noetherian, meaning they satisfy the ascending chain condition on ideals, which leads to many important properties in module theory.
3. Every Dedekind domain is integrally closed in its field of fractions, meaning that any element that is integral over the domain actually lies within it.
4. The class group of a Dedekind domain measures the failure of unique factorization within the ring, indicating how ideals behave relative to one another.
5. Examples of Dedekind domains include the ring of integers in any number field and polynomial rings in one variable over a field.

Review Questions

• How does the property of having every non-zero prime ideal being maximal impact the structure of a Dedekind domain?
  • This property ensures that in a Dedekind domain, any prime ideal corresponds directly to an irreducible element or maximal ideal. This creates a simple structure for factoring ideals since every non-zero proper ideal can be expressed uniquely as a product of these prime ideals. This facilitates understanding how fractional ideals behave and how they interact within the ring.
• Discuss how Dedekind domains relate to the concept of integral extensions and their importance in algebraic number theory.
  • Dedekind domains play a crucial role in defining integral extensions, which are essential for studying algebraic integers. An integral extension occurs when elements are integral over some base ring, and Dedekind domains provide a clear framework for identifying such elements. This connection helps determine properties like whether rings remain integrally closed and ensures that algebraic integers behave well under various operations.
• Evaluate the implications of Dedekind domains being integrally closed on their applications in arithmetic geometry and number theory.
  • The fact that Dedekind domains are integrally closed means they provide a robust setting for working with algebraic varieties and schemes in arithmetic geometry. Being integrally closed allows mathematicians to derive meaningful results about the nature of solutions to polynomial equations over these domains. This characteristic also aids in understanding local behavior near points on varieties and contributes to developing more complex structures like schemes, ultimately impacting the broader landscape of number theory.

Related terms

Fractional Ideal: A fractional ideal is a generalization of an ideal in a ring, allowing for the multiplication by non-zero elements in the field of fractions, used to study properties of Dedekind domains.

Integral Closure: The integral closure of an integral domain is the set of elements in its field of fractions that are roots of monic polynomials with coefficients in the domain, providing insights into integral extensions.

Noetherian Ring: A Noetherian ring is a ring in which every ascending chain of ideals stabilizes, ensuring certain finiteness conditions that are often found in Dedekind domains.