The lying over property refers to a situation in ring theory where, given a ring homomorphism from a local ring to another local ring, any prime ideal in the second ring that lies over a prime ideal in the first ring contains a prime ideal that is the image of the first one. This concept is crucial when discussing how properties of ideals behave under ring extensions and is particularly relevant when understanding integral extensions and the behavior of prime ideals during these transitions.
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The lying over property ensures that prime ideals can be tracked through ring extensions, maintaining some form of correspondence between them.
This property plays a key role in the proof of the Going Up Theorem, establishing how prime ideals behave under integral extensions.
If a local ring is integrally closed, then its lying over property contributes to establishing various properties of its extensions.
Understanding the lying over property can help prove other important results regarding flatness and finite type algebras.
The lying over property also facilitates discussions around minimal primes and height of ideals in extended rings.
Review Questions
How does the lying over property relate to prime ideals during an integral extension?
The lying over property indicates that when you have an integral extension of rings, any prime ideal in the larger ring that lies over a prime ideal in the smaller ring must contain at least one prime ideal that corresponds to it. This means that there's a structured way to understand how these ideals are related, which helps in analyzing their properties and behavior as we move through ring extensions.
Discuss how the lying over property contributes to proving the Going Up Theorem.
The lying over property is fundamental to the Going Up Theorem because it guarantees that for every prime ideal in the original ring, there exists a prime ideal in the extended ring that lies above it. This creates a clear pathway for showing that certain types of elements and ideals behave consistently as you move from one ring to another. Therefore, this property is essential for establishing the existence of primes that can be tracked across integral extensions.
Evaluate the implications of lying over property for integrally closed domains and their extensions.
When dealing with integrally closed domains, the lying over property can lead to significant insights about how these domains behave under extension. Since integrally closed domains have no non-trivial integral elements outside their own structure, this property helps reinforce their stability during extensions. Understanding how prime ideals transition with this property allows mathematicians to conclude things about both height and dimension of ideals in extended rings, providing deeper insights into algebraic geometry and number theory contexts.
An extension of rings where every element of the larger ring is integral over the smaller ring, meaning it satisfies a monic polynomial with coefficients from the smaller ring.
A theorem that states if there is an integral extension of rings, then for any prime ideal in the base ring, there exists a prime ideal in the extended ring lying over it.