Integral Closure

Definition

The integral closure of a ring is the set of elements in its field of fractions that are integral over the ring. This concept is crucial in understanding how certain properties of rings behave under inclusion and extension, particularly regarding their ideals and dimensions. Integral closures help in analyzing how rings can be 'completed' or 'filled out' by adding elements that satisfy polynomial equations with coefficients in the original ring.

5 Must Know Facts For Your Next Test

1. Integral closures can help determine whether a given ring is integrally closed, meaning it contains all elements that are integral over itself.
2. In a Noetherian ring, the integral closure has nice properties regarding finitely generated modules and ideals.
3. For Dedekind domains, the integral closure corresponds to the concept of class groups and plays an important role in algebraic number theory.
4. The process of finding the integral closure is often used in algebraic geometry to study schemes and their properties.
5. Going up and going down theorems provide critical insights into how integral closures behave when moving between rings and their extensions.

Review Questions

• How does understanding the integral closure of a ring enhance our knowledge of its structure and properties?
  • Understanding the integral closure allows us to identify elements that complete the ring in terms of its algebraic relationships. For instance, if a ring is not integrally closed, we can find new elements that can be added to ensure all roots of monic polynomials with coefficients in the ring are included. This insight can reveal important structural information about the ring, such as whether it has unique factorization or certain ideal properties.
• Discuss the implications of the Going Up Theorem in relation to the concept of integral closures.
  • The Going Up Theorem states that if we have an extension of rings, then any prime ideal in the smaller ring can be lifted to a prime ideal in the larger ring under certain conditions. This theorem relates closely to integral closures because it ensures that when we extend our base ring, the prime ideals maintain their structure, allowing us to study integral elements effectively. The preservation of these ideals ensures that we can track which elements remain integral over our original ring when moving to its closure.
• Evaluate how the concept of integral closure interacts with localization and its significance in commutative algebra.
  • The concept of integral closure interacts with localization by allowing us to analyze how local properties affect global behavior. When we localize a ring, we essentially zoom into specific elements and their relationships. Understanding the integral closure helps us see if localized rings maintain properties like being Noetherian or integrally closed. This interaction is significant because it aids in understanding how local conditions can influence broader algebraic structures, which is essential for applications in algebraic geometry and number theory.

Related terms

Integral Element: An element is integral over a ring if it satisfies a monic polynomial equation with coefficients from that ring.

Noetherian Ring: A ring in which every ascending chain of ideals stabilizes, which implies that every ideal is finitely generated.

Ring of Integers: The integral closure of the ring of integers within a number field, consisting of all algebraic integers in that field.