Integral closure of r

Definition

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.

5 Must Know Facts For Your Next Test

1. The integral closure of r can be thought of as capturing all 'nice' elements from the field of fractions that behave well with respect to r.
2. If a ring r is integrally closed, then its integral closure is simply r itself, meaning there are no additional elements to include.
3. In one-dimensional local rings, the integral closure coincides with the normalization of the ring, which plays a role in algebraic geometry.
4. Finding the integral closure can be important for deducing whether a given ring is a Dedekind domain, which has implications for factorization and ideal structure.
5. The process of taking integral closures can lead to extensions that are finitely generated as algebras over r, particularly in cases like polynomial rings.

Review Questions

• How does the concept of integral closure relate to the notion of being integrally closed, and why is this important?
  • Integral closure refers to all elements in the field of fractions that satisfy a monic polynomial over a given ring. A ring being integrally closed means it contains all such elements already within itself. This distinction is crucial because understanding whether a ring is integrally closed can influence its structure and behavior in algebraic contexts, such as when looking at extensions or determining properties like being a Dedekind domain.
• In what ways does finding the integral closure of a ring impact its algebraic properties, particularly concerning extensions?
  • Finding the integral closure can reveal vital information about the algebraic properties of a ring, such as whether it supports unique factorization or behaves well under certain operations. Integral closures often lead to finitely generated algebras, indicating that the extension maintains some structure similar to that of the original ring. Additionally, it allows mathematicians to analyze how extensions interact with ideals and facilitate understanding about normality and other key characteristics.
• Evaluate the significance of the integral closure within algebraic geometry and number theory by discussing its applications and implications.
  • The integral closure plays a pivotal role in both algebraic geometry and number theory by ensuring that rings possess desirable properties such as normality and integrally closedness. In algebraic geometry, working with schemes requires understanding how local rings behave under base changes and extensions; having an integral closure allows for smooth transitions between these structures. In number theory, examining rings through their integral closures helps identify unique factorizations and resolve questions surrounding divisibility and arithmetic properties, enriching our understanding of numerical systems.

Related terms

Integrally Closed Domain: A domain is integrally closed if it contains all of its integral elements over itself. In simpler terms, it means that if an element in its field of fractions is integral over the domain, then that element must already belong to the domain.

Field of Fractions: The field of fractions of an integral domain is the smallest field in which the domain can be embedded, consisting of ratios of its elements. This concept is essential when discussing integral closure as it provides the space where integral elements are considered.

Normal Ring: A normal ring is a ring that is integrally closed in its field of fractions. Normal rings have desirable properties, especially in algebraic geometry and number theory, as they allow for certain kinds of extensions and factorizations.