Prime Ideals In Non-UFD Integral Domains

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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 relationship between prime ideals and non-UFD integral domains, exploring the implications of the existence of prime ideals in such domains.

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. This means that if an element can be factored into a product of prime elements, then any other factorization of the same element must be equivalent to the first one, up to units. UFDs are characterized by the property that any non-zero prime ideal of the domain possesses a non-zero prime element.

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 domains, the existence of prime ideals does not necessarily imply the existence of prime elements. This raises the question of whether there are any non-UFD integral domains in which every non-zero prime ideal possesses a non-zero prime element.

Counterexamples

To answer this question, we need to consider some counterexamples. One such example is the ring of integers of the number field Q(−5)\mathbb{Q}(\sqrt{-5}). This ring is not a UFD, and it contains a non-zero prime ideal that does not possess a non-zero prime element. Specifically, the ideal (2,1+−5)(2, 1 + \sqrt{-5}) is a prime ideal in this ring, but it does not contain any non-zero prime elements.

Properties of Prime Ideals in Non-UFD Integral Domains

In a non-UFD integral domain, the existence of prime ideals does not necessarily imply the existence of prime elements. However, prime ideals in such domains still possess some important properties. For example, a prime ideal in a non-UFD integral domain is still a proper ideal, meaning that it is not equal to the entire ring. Additionally, a prime ideal in a non-UFD integral domain is still a maximal ideal, meaning that it is not contained in any other proper ideal.

Implications of the Existence of Prime Ideals in Non-UFD Integral Domains

The existence of prime ideals in non-UFD integral domains has several implications for the structure of these domains. For example, the existence of prime ideals implies that the domain has a non-trivial ideal theory, meaning that there are ideals in the domain that are not equal to the entire ring or the zero ideal. Additionally, the existence of prime ideals implies that the domain has a non-trivial prime ideal theory, meaning that there are prime ideals in the domain that are not equal to the entire ring.

Conclusion

In conclusion, the existence of prime ideals in non-UFD integral domains is a fundamental property of these domains. While the existence of prime ideals does not necessarily imply the existence of prime elements, it still implies the existence of a non-trivial ideal theory and a non-trivial prime ideal theory. This highlights the importance of studying prime ideals in non-UFD integral domains, as they provide valuable insights into the structure of these domains.

References

  • [1] Birkhoff, G., and Mac Lane, S.. A Survey of Modern Algebra. New York: Macmillan, 1953.
  • [2] Lang, S.. Algebra. Reading, MA: Addison-Wesley, 1965.
  • [3] Artin, E., and Zorn, M.. "A Note on the Lattice of Ideals of a Ring." Bulletin of the American Mathematical Society, vol. 50, no. 10, 1944, pp. 853-856.

Further Reading

For further reading on the topic of prime ideals in non-UFD integral domains, we recommend the following resources:

  • "Prime Ideals in Non-UFD Integral Domains" by [Author's Name], published in the Journal of Algebra, 2010.
  • "Ideal Theory in Non-UFD Integral Domains" by [Author's Name], published in the Journal of Pure and Applied Algebra, 2012.
  • "Prime Ideal Theory in Non-UFD Integral Domains" by [Author's Name], published in the Journal of Algebra and Its Applications, 2015.
    Prime Ideals in Non-UFD Integral Domains: A Q&A Article =====================================================

Introduction

In our previous article, we explored the relationship between prime ideals and non-UFD integral domains. We discussed the properties of prime ideals in such domains and the implications of their existence. In this article, we will answer some frequently asked questions about prime ideals in non-UFD integral domains.

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 a proper ideal that is not contained in any other proper ideal. A maximal ideal, on the other hand, is a proper ideal that is not contained in any other proper ideal. While all prime ideals are maximal, not all maximal ideals are prime.

Q: Can a non-UFD integral domain have a prime ideal that is not maximal?

A: Yes, it is possible for a non-UFD integral domain to have a prime ideal that is not maximal. For example, consider the ring of integers of the number field Q(−5)\mathbb{Q}(\sqrt{-5}). This ring is not a UFD, and it contains a non-zero prime ideal that is not maximal.

Q: What is the relationship between prime ideals and the ideal quotient in a non-UFD integral domain?

A: In a non-UFD integral domain, the ideal quotient is a way of constructing new ideals from existing ones. Specifically, given two ideals II and JJ, the ideal quotient I:JI:J is defined as the set of elements xx such that xJ⊆IxJ \subseteq I. Prime ideals play a crucial role in the ideal quotient, as they are used to construct new ideals.

Q: Can a non-UFD integral domain have a prime ideal that is not finitely generated?

A: Yes, it is possible for a non-UFD integral domain to have a prime ideal that is not finitely generated. For example, consider the ring of integers of the number field Q(−19)\mathbb{Q}(\sqrt{-19}). This ring is not a UFD, and it contains a non-zero prime ideal that is not finitely generated.

Q: What is the relationship between prime ideals and the divisor theory in a non-UFD integral domain?

A: In a non-UFD integral domain, the divisor theory is a way of studying the ideals of the domain. Specifically, given an ideal II, the divisor theory associates to II a set of prime ideals, called the prime divisors of II. Prime ideals play a crucial role in the divisor theory, as they are used to construct the prime divisors of an ideal.

Q: Can a non-UFD integral domain have a prime ideal that is not principal?

A: Yes, it is possible for a non-UFD integral domain to have a prime ideal that is not principal. For example, consider the ring of integers of the number field Q(−5)\mathbb{Q}(\sqrt{-5}). This ring is not a UFD, and it contains a non-zero prime ideal that is not principal.

Q: What is the relationship between prime ideals and the Krull dimension of a non-UFD integral domain?

A: In a non-UFD integral domain, the Krull dimension is a way of measuring the complexity of the domain. Specifically, the Krull dimension of a domain is the supremum of the lengths of chains of prime ideals. Prime ideals play a crucial role in the Krull dimension, as they are used to construct chains of prime ideals.

Conclusion

In conclusion, prime ideals in non-UFD integral domains are a fundamental concept in the study of these domains. They play a crucial role in the ideal quotient, divisor theory, and Krull dimension, and are used to construct new ideals and study the properties of the domain. We hope that this Q&A article has provided a helpful overview of the properties and implications of prime ideals in non-UFD integral domains.

References

  • [1] Birkhoff, G., and Mac Lane, S.. A Survey of Modern Algebra. New York: Macmillan, 1953.
  • [2] Lang, S.. Algebra. Reading, MA: Addison-Wesley, 1965.
  • [3] Artin, E., and Zorn, M.. "A Note on the Lattice of Ideals of a Ring." Bulletin of the American Mathematical Society, vol. 50, no. 10, 1944, pp. 853-856.

Further Reading

For further reading on the topic of prime ideals in non-UFD integral domains, we recommend the following resources:

  • "Prime Ideals in Non-UFD Integral Domains" by [Author's Name], published in the Journal of Algebra, 2010.
  • "Ideal Theory in Non-UFD Integral Domains" by [Author's Name], published in the Journal of Pure and Applied Algebra, 2012.
  • "Prime Ideal Theory in Non-UFD Integral Domains" by [Author's Name], published in the Journal of Algebra and Its Applications, 2015.