Zero Divisors and Prime Ideals

Oct 13, 2025 by windailys.com 142 views

Introduction

In the realm of commutative algebra, the concept of flatness plays a crucial role in understanding the properties of ideals within a ring. A flat ideal is one that preserves the exactness of sequences of modules, making it a fundamental tool in algebraic geometry and representation theory. However, when dealing with reduced rings, which have all prime ideals maximal, the landscape becomes more complex. In this article, we will delve into the question of whether an ideal within a set of zero divisors of a reduced ring can be flat.

Background and Notation

Let $A$ be a reduced ring, meaning that it has no nonzero nilpotent elements. We assume that $A$ is non-Noetherian, with infinitely many prime ideals $\mathfrak{P}_i$ , where $i\in I$ . Our goal is to investigate the flatness of an ideal $\mathfrak{a}\subset\mathfrak{P}_i$ within this set of zero divisors.

Zero Divisors and Prime Ideals

A zero divisor in a ring $A$ is an element $a\in A$ such that there exists a nonzero element $b\in A$ with $ab=0$ . In a reduced ring, the zero divisors are precisely the elements that are contained in some prime ideal. Since $A$ has infinitely many prime ideals, we can consider the set of zero divisors as a collection of prime ideals.

Flatness and the Tor Functor

The flatness of an ideal $\mathfrak{a}$ is determined by the behavior of the Tor functor, which measures the failure of exactness in a sequence of modules. Specifically, an ideal $\mathfrak{a}$ is flat if and only if the Tor functor $\text{Tor}_1^A(M,\mathfrak{a})=0$ for all $A$ -modules $M$ . This condition ensures that the ideal $\mathfrak{a}$ preserves the exactness of sequences of modules.

Reduced Rings and the Tor Functor

In a reduced ring $A$ , the Tor functor is particularly well-behaved. Since $A$ has no nonzero nilpotent elements, the Tor functor is always zero for modules of finite length. However, when dealing with modules of infinite length, the Tor functor can be non-zero. In this case, the flatness of an ideal $\mathfrak{a}$ is determined by the behavior of the Tor functor for modules of infinite length.

The Main Question

Given an ideal $\mathfrak{a}\subset\mathfrak{P}_i$ within a set of zero divisors of a reduced ring $A$ , can we determine whether $\mathfrak{a}$ is flat? This question is central to our investigation, and we will explore various approaches to answering it.

Approach 1: Using the Tor Functor

One possible approach to determining the flatness of an ideal $\mathfrak{a}$ is to use the Tor functor. Specifically, we can consider the sequence of modules

0\to M\to N\to \mathfrak{a}\to 0

where $M$ and $N$ are $A$ -modules. If the Tor functor $\text{Tor}_1^A(M,\mathfrak{a})=0$ , then the ideal $\mathfrak{a}$ is flat. However, if the Tor functor is non-zero, then $\mathfrak{a}$ may not be flat.

Approach 2: Using the Prime Ideal Spectrum

Another approach to determining the flatness of an ideal $\mathfrak{a}$ is to use the prime ideal spectrum of the ring $A$ . Specifically, we can consider the set of prime ideals $\mathfrak{P}_i$ that contain the ideal $\mathfrak{a}$ . If the ideal $\mathfrak{a}$ is contained in a single prime ideal, then it is flat. However, if $\mathfrak{a}$ is contained in multiple prime ideals, then its flatness is more difficult to determine.

Approach 3: Using the Ideal Quotient

A third approach to determining the flatness of an ideal $\mathfrak{a}$ is to use the ideal quotient. Specifically, we can consider the ideal quotient $\mathfrak{a}:\mathfrak{b}$ , where $\mathfrak{b}$ is another ideal in the ring $A$ . If the ideal quotient is flat, then the ideal $\mathfrak{a}$ is also flat.

Conclusion

In conclusion, the flatness of an ideal $\mathfrak{a}$ within a set of zero divisors of a reduced ring $A$ is a complex question that requires careful consideration of various approaches. Using the Tor functor, the prime ideal spectrum, and the ideal quotient, we can determine whether an ideal $\mathfrak{a}$ is flat. However, the answer to this question is not always straightforward, and further research is needed to fully understand the properties of flat ideals in reduced rings.

References

[1] Bourbaki, N. (1961). Commutative Algebra. Hermann.
[2] Cartan, H., & Eilenberg, S. (1956). Homological Algebra. Princeton University Press.
[3] Matsumura, H. (1980). Commutative Ring Theory. Cambridge University Press.

Future Directions

Future research directions in this area include:

Investigating the flatness of ideals in reduced rings with infinitely many prime ideals.
Developing new approaches to determining the flatness of ideals using the Tor functor, the prime ideal spectrum, and the ideal quotient.
Exploring the relationship between flat ideals and other algebraic properties, such as Noetherianity and Artinianity.

Introduction

In our previous article, we explored the concept of flatness of an ideal inside a set of zero divisors of a reduced ring. We discussed various approaches to determining the flatness of an ideal, including using the Tor functor, the prime ideal spectrum, and the ideal quotient. In this article, we will answer some frequently asked questions about flat ideals in reduced rings.

Q: What is the significance of flat ideals in reduced rings?

A: Flat ideals in reduced rings are significant because they preserve the exactness of sequences of modules. This property makes them useful in algebraic geometry and representation theory, where exactness is crucial for understanding the behavior of modules and their properties.

Q: How do I determine if an ideal is flat in a reduced ring?

A: To determine if an ideal is flat in a reduced ring, you can use the Tor functor, the prime ideal spectrum, or the ideal quotient. Each of these approaches has its own strengths and weaknesses, and the choice of approach depends on the specific properties of the ideal and the ring.

Q: What is the relationship between flat ideals and Noetherianity?

A: Flat ideals in reduced rings are not necessarily Noetherian. In fact, many flat ideals in reduced rings are non-Noetherian. However, the relationship between flat ideals and Noetherianity is complex, and further research is needed to fully understand this connection.

Q: Can I use the Tor functor to determine the flatness of an ideal?

A: Yes, the Tor functor can be used to determine the flatness of an ideal. Specifically, if the Tor functor is zero for all modules, then the ideal is flat. However, if the Tor functor is non-zero for some modules, then the ideal may not be flat.

Q: What is the significance of the prime ideal spectrum in determining flatness?

A: The prime ideal spectrum is significant in determining flatness because it provides a way to understand the behavior of ideals in a reduced ring. Specifically, if an ideal is contained in a single prime ideal, then it is flat. However, if the ideal is contained in multiple prime ideals, then its flatness is more difficult to determine.

Q: Can I use the ideal quotient to determine the flatness of an ideal?

A: Yes, the ideal quotient can be used to determine the flatness of an ideal. Specifically, if the ideal quotient is flat, then the ideal is also flat. However, if the ideal quotient is non-flat, then the ideal may not be flat.

Q: What are some common mistakes to avoid when working with flat ideals in reduced rings?

A: Some common mistakes to avoid when working with flat ideals in reduced rings include:

Assuming that all ideals in a reduced ring are flat.
Failing to use the Tor functor, the prime ideal spectrum, or the ideal quotient to determine flatness.
Ignoring the relationship between flat ideals and Noetherianity.
Not considering the properties of the ideal and the ring when determining flatness.

Conclusion

In conclusion, the flatness of an ideal inside a set of zero divisors of a reduced ring is a complex question that requires careful consideration of various approaches. By understanding the significance of flat ideals, determining flatness using the Tor functor, the prime ideal spectrum, and the ideal quotient, and avoiding common mistakes, we can gain a deeper insight into the properties of flat ideals in reduced rings.

References

[1] Bourbaki, N. (1961). Commutative Algebra. Hermann.
[2] Cartan, H., & Eilenberg, S. (1956). Homological Algebra. Princeton University Press.
[3] Matsumura, H. (1980). Commutative Ring Theory. Cambridge University Press.

Future Directions

Future research directions in this area include:

Investigating the flatness of ideals in reduced rings with infinitely many prime ideals.
Developing new approaches to determining the flatness of ideals using the Tor functor, the prime ideal spectrum, and the ideal quotient.
Exploring the relationship between flat ideals and other algebraic properties, such as Noetherianity and Artinianity.

By advancing our understanding of flat ideals in reduced rings, we can gain a deeper insight into the properties of these rings and their applications in algebraic geometry and representation theory.