A Prime Ideal P Contains the Ideal I iff P Contains One of the Associated Primes of I

Oct 13, 2025 by leolavanderia.com.br 74 views

A Prime Ideal P Contains the Ideal I iff P Contains One of the Associated Primes of I

In the realm of Commutative Algebra, the study of prime ideals and their relationship with other ideals is a fundamental concept. A prime ideal is a proper ideal in a ring that satisfies the property that if the product of two elements is in the ideal, then at least one of the elements must be in the ideal. In this article, we will explore the relationship between a prime ideal P and an ideal I, specifically showing that P contains I if and only if P contains one of the associated primes of I.

Let R be a Noetherian ring, which means that every non-empty set of ideals in R has a maximal element. This is a crucial property that will be used throughout our discussion. Let I be a proper ideal in R, meaning that I is not equal to R. We denote the set of associated primes of I as Ass(I), which consists of all prime ideals P in R such that P is the annihilator of some element in the module R/I.

To show that P contains I if and only if P contains one of the associated primes of I, we will first prove the forward direction. Suppose that P contains I. We need to show that P contains one of the associated primes of I.

Step 1: P contains the annihilator of some element in R/I

Since P contains I, we know that P is a proper ideal in R. Let x be an element in R/I such that the annihilator of x is a prime ideal in R. We claim that P contains the annihilator of x.

Proof: Let Q be the annihilator of x. Since Q is a prime ideal, we know that Q is a minimal prime ideal in R. By the definition of a minimal prime ideal, we have that Q is the annihilator of some element in R. Let y be an element in R such that Q is the annihilator of y. Since P contains I, we know that P contains the annihilator of y. Therefore, P contains Q, which is the annihilator of x.

Step 2: P contains one of the associated primes of I

Since P contains the annihilator of x, we know that P is one of the associated primes of I. Therefore, P contains one of the associated primes of I.

Next, we will prove the backward direction. Suppose that P contains one of the associated primes of I. We need to show that P contains I.

Step 1: P contains the annihilator of some element in R/I

Let Q be one of the associated primes of I. Since P contains Q, we know that P contains the annihilator of some element in R/I.

Proof: Let x be an element in R/I such that Q is the annihilator of x. Since Q is a prime ideal, we know that Q is a minimal prime ideal in R. By the definition of a minimal prime ideal, we have that Q is the annihilator of some element in R. Let y be an element in R such that Q is the annihilator of y. Since P contains Q, we know that P contains the annihilator of y. Therefore, P contains the annihilator of x.

Step 2: P contains I

Since P contains the annihilator of x, we know that P contains I.

In conclusion, we have shown that a prime ideal P contains the ideal I if and only if P contains one of the associated primes of I. This result is a fundamental property of prime ideals and their relationship with other ideals in a Noetherian ring.

[1] Eisenbud, D. (1995). Commutative Algebra with a View Toward Algebraic Geometry. Springer-Verlag.
[2] Matsumura, H. (1989). Commutative Ring Theory. Cambridge University Press.

For further reading on this topic, we recommend the following:

[1] Commutative Algebra by David Eisenbud
[2] Commutative Ring Theory by Hideyuki Matsumura

Associated prime: A prime ideal P in R such that P is the annihilator of some element in the module R/I.
Minimal prime ideal: A prime ideal Q in R such that Q is the annihilator of some element in R.
Noetherian ring: A ring R such that every non-empty set of ideals in R has a maximal element.
Prime ideal: A proper ideal P in R such that if the product of two elements is in the ideal, then at least one of the elements must be in the ideal.
A Prime Ideal P Contains the Ideal I iff P Contains One of the Associated Primes of I: Q&A

In our previous article, we explored the relationship between a prime ideal P and an ideal I, specifically showing that P contains I if and only if P contains one of the associated primes of I. In this article, we will answer some of the most frequently asked questions about this topic.

Q: What is the significance of associated primes in this context?

A: Associated primes play a crucial role in this context because they are the prime ideals that are "closest" to the ideal I. In other words, they are the prime ideals that are most closely related to I. By showing that P contains I if and only if P contains one of the associated primes of I, we are highlighting the importance of associated primes in understanding the relationship between prime ideals and ideals.

Q: What is the difference between a prime ideal and an associated prime?

A: A prime ideal is a proper ideal in a ring that satisfies the property that if the product of two elements is in the ideal, then at least one of the elements must be in the ideal. An associated prime, on the other hand, is a prime ideal that is the annihilator of some element in the module R/I. In other words, an associated prime is a prime ideal that is "related" to the ideal I.

Q: How do we find the associated primes of an ideal?

A: To find the associated primes of an ideal I, we need to find the prime ideals P in R such that P is the annihilator of some element in the module R/I. This can be done using various methods, such as using the primary decomposition of I or using the concept of the "associated graded ring" of I.

Q: What is the relationship between associated primes and the primary decomposition of an ideal?

A: The primary decomposition of an ideal I is a way of expressing I as the intersection of primary ideals. Associated primes play a crucial role in this decomposition, as they are the prime ideals that are "closest" to the primary ideals. In other words, the associated primes of I are the prime ideals that are most closely related to the primary ideals in the primary decomposition of I.

Q: How do we use associated primes to understand the relationship between prime ideals and ideals?

A: Associated primes can be used to understand the relationship between prime ideals and ideals in several ways. For example, they can be used to show that a prime ideal P contains an ideal I if and only if P contains one of the associated primes of I, as we did in our previous article. They can also be used to show that the associated primes of an ideal I are the prime ideals that are "most closely related" to I.

Q: What are some of the applications of associated primes in algebraic geometry and commutative algebra?

A: Associated primes have many applications in algebraic geometry and commutative algebra. For example, they can be used to study the properties of singularities and to understand the relationship between prime ideals and ideals. They can also be used to study the properties of modules and to understand the relationship between modules and ideals.

Q: What are some of the open problems in the study of associated primes?

A: There are many open problems in the study of associated primes. For example, it is still not known whether the associated primes of an ideal I are always the prime ideals that are "most closely related" to I. It is also not known whether the associated primes of an ideal I are always the prime ideals that are "closest" to the primary ideals in the primary decomposition of I.

In conclusion, associated primes play a crucial role in understanding the relationship between prime ideals and ideals. They can be used to show that a prime ideal P contains an ideal I if and only if P contains one of the associated primes of I, and they can be used to study the properties of singularities and to understand the relationship between prime ideals and ideals. There are many open problems in the study of associated primes, and further research is needed to fully understand their significance.

[1] Eisenbud, D. (1995). Commutative Algebra with a View Toward Algebraic Geometry. Springer-Verlag.
[2] Matsumura, H. (1989). Commutative Ring Theory. Cambridge University Press.

For further reading on this topic, we recommend the following:

[1] Commutative Algebra by David Eisenbud
[2] Commutative Ring Theory by Hideyuki Matsumura

Associated prime: A prime ideal P in R such that P is the annihilator of some element in the module R/I.
Minimal prime ideal: A prime ideal Q in R such that Q is the annihilator of some element in R.
Noetherian ring: A ring R such that every non-empty set of ideals in R has a maximal element.
Prime ideal: A proper ideal P in R such that if the product of two elements is in the ideal, then at least one of the elements must be in the ideal.