An integral extension is a type of ring extension where every element of the extended ring is integral over the base ring, meaning each element satisfies a monic polynomial with coefficients in the base ring. This concept connects deeply with integral elements and how they interact between rings, establishing foundational properties that influence the behavior of ideals and prime ideals during extensions.
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Integral extensions preserve many properties of rings, such as Noetherian and integrally closed properties.
If A is a ring and B is an integral extension of A, then B is also a finitely generated A-module.
The integral closure of a ring in its field of fractions is an example of an integral extension.
If a ring is Noetherian, then any integral extension is also Noetherian.
Integral extensions are often used to study local properties of rings, especially in algebraic geometry.
Review Questions
How does being an integral extension affect the properties of the rings involved?
Being an integral extension affects many properties of the rings, such as preserving Noetherian conditions. If you have a ring A and an integral extension B, then B inherits certain characteristics from A, such as being Noetherian or integrally closed. This is significant because it allows mathematicians to apply results known for the base ring to the extended ring.
What role do integral elements play in understanding the structure of an integral extension?
Integral elements are central to defining an integral extension, as each element of the extension must satisfy a monic polynomial from the base ring. By examining these integral elements, one can study how ideals behave under extensions and how properties like localization are preserved. This analysis is crucial for understanding the overall structure and function of both rings involved.
Evaluate the implications of the Going Up Theorem within the context of integral extensions and its relevance to prime ideals.
The Going Up Theorem has profound implications in integral extensions as it guarantees that if you have a prime ideal in the base ring, there will be corresponding prime ideals in the extension that lie over it. This is particularly relevant for understanding how prime ideals interact and how they may split or remain intact under extensions. The theorem not only aids in exploring the structure of rings but also connects to broader concepts in algebraic geometry where understanding prime ideals is vital.
A broader class of extensions where each element of the extended field satisfies a polynomial equation with coefficients in the base field, not necessarily monic.
A statement about the behavior of prime ideals in an integral extension, asserting that if a prime ideal lies over another, then there exists a prime ideal in the extension lying over it.