Integral Extensions

Definition

Integral extensions are a type of ring extension where each element of the extension ring is integral over the base ring. This means that for any element in the extension, there exists a monic polynomial with coefficients from the base ring such that the element is a root of that polynomial. Understanding integral extensions is crucial for exploring integrally closed domains, which are rings that coincide with their integral closures, and for studying properties related to factorization and local rings.

5 Must Know Facts For Your Next Test

1. An integral extension preserves properties such as Noetherianity; if the base ring is Noetherian, then so is the integral extension.
2. If an extension is finitely generated as an algebra over its base ring, then it is integral if it is integrally closed.
3. Integral extensions provide a way to study roots of polynomials and their behavior in different rings, particularly in algebraic geometry.
4. Every finitely generated algebra over a field is integral over its subring, which links algebraic and geometric properties.
5. An important property of integral extensions is that they can be characterized through valuation rings, where each nonzero prime ideal corresponds to an element that can be made integral.

Review Questions

• How does the property of being integral relate to the structure of integral extensions?
  • Being integral means that elements of the extension satisfy monic polynomials with coefficients from the base ring, leading to various structural consequences. For instance, this property ensures that the extension behaves nicely in terms of ideal structures and factorization. When you have an integral extension, any prime ideal in the base ring remains prime in the extended ring, reflecting a strong compatibility between both structures.
• What are the implications of integral extensions on Noetherian rings, and why are these implications significant?
  • Integral extensions have significant implications on Noetherian rings because if you start with a Noetherian base ring, any integral extension will also be Noetherian. This matters because Noetherian rings possess desirable properties like finite generation of ideals and good behavior under localization. Thus, understanding how these properties carry over to integral extensions helps in broader algebraic studies, especially when dealing with schemes and varieties.
• Evaluate how integral extensions influence the relationship between local rings and their global counterparts.
  • Integral extensions significantly influence how local rings relate to their global counterparts by revealing information about singularities and regularity conditions. When you look at a local ring and consider its integral closure, it reflects how localized behavior can inform global geometric properties. This evaluation shows that understanding integral extensions not only aids in grasping algebraic structures but also plays a key role in connecting algebra with geometry through concepts like normalization.

Related terms

Integral Closure:

The integral closure of a ring is the set of elements that are integral over that ring, forming the smallest integrally closed domain containing it.

Noetherian Ring:

A Noetherian ring is a ring in which every ascending chain of ideals stabilizes, playing an essential role in the study of integral extensions.

Localization:

Localization is the process of turning a ring into a local ring by allowing division by a subset of elements, which helps in studying properties of integral extensions in a more focused manner.