Prime Ideals In Non-UFD Integral Domains

May 2, 2025 by ADMIN 41 views

**Prime Ideals in Non-UFD Integral Domains**

Introduction

In the realm of abstract algebra, particularly in ring theory, the concept of prime ideals plays a crucial role in understanding the structure of integral domains. An integral domain is a commutative ring with unity that has no zero divisors. However, not all integral domains possess the unique factorization property, which is a fundamental characteristic of unique factorization domains (UFDs). In this article, we will delve into the world of non-UFD integral domains and explore the properties of prime ideals in such rings.

Unique Factorization Domains (UFDs)

A unique factorization domain is an integral domain in which every non-zero element can be expressed as a product of prime elements in a unique way, up to units. In other words, if an element can be factored into a product of prime elements, then any other factorization of the same element must be identical, except for the order of the factors and the presence of units. The existence of unique factorization is a key property of UFDs, and it has far-reaching implications for the structure of these rings.

Non-UFD Integral Domains

Not all integral domains are UFDs. In fact, there are many examples of non-UFD integral domains, such as the ring of integers of a number field that is not a UFD. In such rings, the property of unique factorization fails, and the factorization of elements can be quite complex. In this article, we will focus on the properties of prime ideals in non-UFD integral domains.

Prime Ideals in Non-UFD Integral Domains

A prime ideal in a ring is an ideal that satisfies the following property: if the product of two elements in the ring is in the ideal, then at least one of the elements must be in the ideal. In other words, a prime ideal is an ideal that is "prime" with respect to multiplication. In a non-UFD integral domain, the existence of prime ideals is still guaranteed, but the properties of these ideals can be quite different from those in UFDs.

The Relationship Between Prime Ideals and Prime Elements

In a UFD, any nonzero prime ideal possesses a nonzero prime element. This means that if an element is in a prime ideal, then it must be a prime element itself. However, in a non-UFD integral domain, this relationship between prime ideals and prime elements can be more complex. In fact, there are examples of non-UFD integral domains in which a prime ideal does not contain a nonzero prime element.

Examples of Non-UFD Integral Domains

One classic example of a non-UFD integral domain is the ring of integers of the number field $\mathbb{Q}(\sqrt{-5})$ . This ring is not a UFD, and it contains a prime ideal that does not contain a nonzero prime element. Specifically, the ideal $(2, 1 + \sqrt{-5})$ is a prime ideal in this ring, but it does not contain a nonzero prime element.

Properties of Prime Ideals in Non-UFD Integral Domains

In a non-UFD integral domain, the properties of prime ideals can be quite different from those in UFDs. For example, a prime ideal in a non-UFD integral domain may not be a maximal ideal, even if it is a prime ideal in the classical sense. Additionally, the factorization of elements in a prime ideal can be quite complex, and it may not be possible to express an element as a product of prime elements in a unique way.

Conclusion

In conclusion, the properties of prime ideals in non-UFD integral domains are quite different from those in UFDs. While the existence of prime ideals is still guaranteed in non-UFD integral domains, the properties of these ideals can be quite complex. In particular, a prime ideal in a non-UFD integral domain may not contain a nonzero prime element, and the factorization of elements in a prime ideal can be quite complex. Further research is needed to fully understand the properties of prime ideals in non-UFD integral domains.

References

[1] Bourbaki, N. (1961). Commutative Algebra. Addison-Wesley.
[2] Eisenbud, D. (1995). Commutative Algebra with a View Toward Algebraic Geometry. Springer-Verlag.
[3] Lang, S. (1993). Algebra. Addison-Wesley.

Q: What is the difference between a prime ideal and a maximal ideal in a non-UFD integral domain?

A: In a non-UFD integral domain, a prime ideal is an ideal that satisfies the property that if the product of two elements in the ring is in the ideal, then at least one of the elements must be in the ideal. A maximal ideal, on the other hand, is an ideal that is not contained in any other proper ideal. In a non-UFD integral domain, a prime ideal may not be a maximal ideal, even if it is a prime ideal in the classical sense.

Q: Can a prime ideal in a non-UFD integral domain contain a nonzero prime element?

A: No, a prime ideal in a non-UFD integral domain may not contain a nonzero prime element. In fact, there are examples of non-UFD integral domains in which a prime ideal does not contain a nonzero prime element.

Q: What is an example of a non-UFD integral domain in which a prime ideal does not contain a nonzero prime element?

A: One classic example of a non-UFD integral domain is the ring of integers of the number field $\mathbb{Q}(\sqrt{-5})$ . This ring is not a UFD, and it contains a prime ideal that does not contain a nonzero prime element. Specifically, the ideal $(2, 1 + \sqrt{-5})$ is a prime ideal in this ring, but it does not contain a nonzero prime element.

Q: How does the factorization of elements in a prime ideal differ from the factorization of elements in a UFD?

A: In a UFD, the factorization of elements is unique, up to units. In a non-UFD integral domain, the factorization of elements in a prime ideal can be quite complex, and it may not be possible to express an element as a product of prime elements in a unique way.

Q: Can a prime ideal in a non-UFD integral domain be a maximal ideal?

A: Yes, a prime ideal in a non-UFD integral domain can be a maximal ideal. However, this is not always the case, and the relationship between prime ideals and maximal ideals in non-UFD integral domains can be quite complex.

Q: What is the significance of prime ideals in non-UFD integral domains?

A: Prime ideals in non-UFD integral domains play a crucial role in understanding the structure of these rings. They provide a way to study the factorization of elements and the properties of ideals in non-UFD integral domains.

Q: How do prime ideals in non-UFD integral domains relate to the concept of unique factorization?

A: Prime ideals in non-UFD integral domains are closely related to the concept of unique factorization. In a UFD, any nonzero prime ideal possesses a nonzero prime element, which is a key property of UFDs. In a non-UFD integral domain, this relationship between prime ideals and prime elements can be more complex.

Q: Can a non-UFD integral domain be a UFD if it has a prime ideal that a nonzero prime element?

A: No, a non-UFD integral domain cannot be a UFD if it has a prime ideal that does not contain a nonzero prime element. However, if a non-UFD integral domain has a prime ideal that contains a nonzero prime element, it may still be possible for the ring to be a UFD.

Q: What are some open questions in the study of prime ideals in non-UFD integral domains?

A: There are many open questions in the study of prime ideals in non-UFD integral domains. Some of these questions include:

What are the necessary and sufficient conditions for a prime ideal in a non-UFD integral domain to contain a nonzero prime element?
How do prime ideals in non-UFD integral domains relate to the concept of unique factorization?
Can a non-UFD integral domain be a UFD if it has a prime ideal that contains a nonzero prime element?

These are just a few examples of the many open questions in the study of prime ideals in non-UFD integral domains. Further research is needed to fully understand the properties of prime ideals in these rings.