Integral Closure

Definition

Integral closure refers to the set of elements in a ring that are integral over that ring, meaning they satisfy a polynomial equation with coefficients in that ring. This concept helps in understanding how certain subrings can be extended to encompass all elements that behave nicely with respect to the original ring, especially in relation to ideals and primary decomposition.

5 Must Know Facts For Your Next Test

1. The integral closure of a ring can often be seen as a way to complete the ring by including all elements that are roots of polynomials with coefficients from the original ring.
2. If a ring is integrally closed, then it contains all its integral elements, making it a useful property for determining the structure of various algebraic objects.
3. In primary decomposition, integral closure helps identify associated primes by analyzing how elements behave in relation to certain ideals.
4. The process of finding the integral closure is closely tied to localization and completion techniques, which help in dealing with local properties of rings.
5. Integral closure can be used to show that every Noetherian domain has a well-defined integral closure, which is essential in algebraic geometry.

Review Questions

• How does the concept of integral closure relate to the structure of primary ideals and their decomposition?
  • Integral closure plays a key role in understanding primary ideals and their decomposition by allowing us to identify which elements are integral over the given ideal. When working with primary decompositions, examining the integral closure helps in determining the associated primes because it captures all elements that can be expressed as roots of polynomials tied to those ideals. This relationship facilitates the classification and analysis of the ring's structure and its prime spectrum.
• Discuss why integral closure is important for proving that certain rings, like Dedekind domains, have nice properties related to their prime ideals.
  • Integral closure is crucial for Dedekind domains because it ensures that every non-zero proper ideal can be uniquely factored into prime ideals. This property is directly connected to the concept of being integrally closed since Dedekind domains contain all integral elements over themselves. As a result, studying the integral closure allows mathematicians to establish deeper insights into how these rings behave under factorization and their overall structure regarding ideals.
• Evaluate the implications of an integral closure being different from its original ring when considering algebraic geometry and schemes.
  • When an integral closure differs from its original ring, it has significant implications for algebraic geometry and schemes. This difference indicates that there are additional algebraic structures or points that exist outside the original configuration, potentially altering the geometric properties of a scheme. Understanding this discrepancy can lead to richer insights about singularities, intersections, and morphisms between varieties, revealing how essential components interact within the broader context of algebraic varieties and their associated schemes.

Related terms

Integral Element: An element of a ring that satisfies a monic polynomial equation with coefficients in another ring.

Dedekind Domain: A type of integral domain where every non-zero proper ideal can be uniquely factored into prime ideals, leading to well-behaved integral closures.

Primary Ideal:

An ideal such that whenever the product of two elements belongs to it, at least one of the elements is contained in the ideal, crucial for understanding the structure of rings.