Prufer Domains

Definition

A Prufer domain is an integral domain in which every finitely generated ideal is projective. This property leads to a nice structure regarding the ideals in the domain, allowing for unique factorizations of elements in certain contexts. The concept of Prufer domains connects closely with Dedekind domains, as every Dedekind domain is a Prufer domain, but not all Prufer domains are Dedekind domains, highlighting important distinctions in their structural properties.

5 Must Know Facts For Your Next Test

1. Prufer domains ensure that every finitely generated ideal behaves nicely, allowing us to work with them more easily compared to general integral domains.
2. In a Prufer domain, any non-zero element can be expressed as a product of prime elements, which relates to the concept of unique factorization in rings.
3. Every Prufer domain is integrally closed in its field of fractions, which means that if an element from the field satisfies a monic polynomial with coefficients in the domain, it must already be in the domain.
4. A key example of a Prufer domain is the ring of all finitely generated projective modules over a Noetherian ring.
5. The connection between Prufer domains and Dedekind domains is significant since while all Dedekind domains are Prufer domains, there exist Prufer domains that do not meet all the criteria to be classified as Dedekind.

Review Questions

• How does being a Prufer domain influence the structure of finitely generated ideals within that domain?
  • Being a Prufer domain means that every finitely generated ideal is projective. This influences the structure by ensuring that these ideals have desirable properties such as being locally free and allowing for certain types of decomposition. Consequently, it allows mathematicians to understand and work with these ideals more effectively compared to other types of integral domains.
• Compare and contrast the properties of Prufer domains and Dedekind domains, focusing on their ideal structures.
  • Both Prufer domains and Dedekind domains share important structural properties regarding their ideals. However, while every Dedekind domain has the characteristic that every non-zero prime ideal is maximal, this is not true for all Prufer domains. In fact, Prufer domains can have more flexible ideal structures since they focus on projective qualities rather than strict maximality conditions, leading to a broader class of examples beyond just Dedekind domains.
• Evaluate how the concept of fractional ideals relates to Prufer domains and their ideal structure.
  • Fractional ideals extend the notion of traditional ideals in integral domains by allowing for elements outside the ring itself, which can provide insights into divisibility and factorization. In the context of Prufer domains, fractional ideals help illustrate how unique factorization can manifest when examining finitely generated ideals. This relationship enriches our understanding of how these ideals function within Prufer domains and shows the overlap between abstract algebraic concepts like projectivity and practical applications in factorization theory.

Related terms

Dedekind Domains: A Dedekind domain is a specific type of integral domain where every non-zero prime ideal is maximal, and all ideals can be uniquely factored into products of prime ideals.

Projective Modules:

A projective module is a type of module that satisfies the property that every surjective module homomorphism onto it can be lifted to a module homomorphism from the original module.

Fractional Ideals:

Fractional ideals are generalizations of ideals in an integral domain that allow for elements that may not belong to the ring itself, often used to study divisibility and unique factorization.