Meaning Of both Sides Collapse And The Correct Map For A Past MSE Post Concerning Localization Of Quotient Rings.

Introduction

In the realm of abstract algebra, particularly in ring theory, the concept of localization plays a crucial role in understanding the properties of rings and their quotient rings. Localization is a process of creating a new ring from an existing one by inverting certain elements, which helps to study the behavior of the original ring in a more refined manner. In this context, the concept of "both sides collapse" arises when considering the localization of quotient rings. This phenomenon has been discussed in a past Mathematics Stack Exchange (MSE) post, where Arturo Magidin provided an insightful answer. In this article, we will delve into the meaning of "both sides collapse" and the correct map for a past MSE post concerning localization of quotient rings.

Background

According to the answer provided by Arturo Magidin for the MSE post "What do elements in a localized quotient ring look like?", we are given a ring $R$ and a prime ideal $P$. The localization of $R$ at $P$, denoted by $R_P$, is defined as the set of equivalence classes of fractions $\frac{a}{s}$, where $a \in R$ and $s \in R \setminus P$. The equivalence relation is defined as $\frac{a}{s} \sim \frac{b}{t}$ if and only if there exists $u \in R \setminus P$ such that $u(at - bs) = 0$. The operations of addition and multiplication are defined as $\frac{a}{s} + \frac{b}{t} = \frac{at + bs}{st}$ and $\frac{a}{s} \cdot \frac{b}{t} = \frac{ab}{st}$, respectively.

The Concept of "Both Sides Collapse"

In the context of localization of quotient rings, the concept of "both sides collapse" refers to a situation where the localization of a quotient ring at a prime ideal $P$ results in a ring that is isomorphic to the localization of the original ring $R$ at $P$. In other words, both sides of the quotient ring collapse to the same ring. This phenomenon can be understood by considering the following commutative diagram:

$$\begin{CD} R @>>> R_P \\ @VVV @VVV \\ R/I @>>> (R/I)_P \end{CD}$$

where $I$ is an ideal of $R$ and $P$ is a prime ideal of $R$. The map $R \to R_P$ is the localization map, and the map $R/I \to (R/I)_P$ is the localization map of the quotient ring. If both sides collapse, then the map $R \to R_P$ is an isomorphism, and the map $R/I \to (R/I)_P$ is also an isomorphism.

The Correct Map for a Past MSE Post

In the past MSE post, Arturo Magidin provided an answer to the question "What do elements in a localized quotient ring look like?". The answer involves a map from the localized quotient ring to the localization of the original ring. To understand the correct map, we need to consider the following commutative diagram:

$$\begin{CD} R @>>> R_P \\ @VVV @VVV \\ R/I @>>> (R/I)_P \end{CD}$$

The map $R \to R_P$ is the localization map, and the map $R/I \to (R/I)_P$ is the localization map of the quotient ring. The correct map from the localized quotient ring to the localization of the original ring is given by the following formula:

$$\frac{a + I}{s + I} \mapsto \frac{a}{s}$$

where $a \in R$ and $s \in R \setminus P$. This map is well-defined and is an isomorphism if and only if both sides collapse.

Example

To illustrate the concept of "both sides collapse" and the correct map, let's consider an example. Suppose we have a ring $R = \mathbb{Z}$ and a prime ideal $P = (2)$. We can define a quotient ring $R/I = \mathbb{Z}/(2)$, where $I = (2)$. The localization of $R$ at $P$ is given by $R_P = \mathbb{Z}_2$, and the localization of the quotient ring is given by $(R/I)_P = (\mathbb{Z}/(2))_2$. If we consider the map from the localized quotient ring to the localization of the original ring, we get:

$$\frac{1 + (2)}{1 + (2)} \mapsto \frac{1}{1}$$

This map is an isomorphism, and both sides collapse.

Conclusion

In conclusion, the concept of "both sides collapse" and the correct map for a past MSE post concerning localization of quotient rings are essential tools in understanding the properties of rings and their quotient rings. By considering the localization of a quotient ring at a prime ideal, we can study the behavior of the original ring in a more refined manner. The correct map from the localized quotient ring to the localization of the original ring is given by a well-defined formula, and it is an isomorphism if and only if both sides collapse. This phenomenon has been illustrated through an example, and it provides a deeper understanding of the properties of rings and their quotient rings.

References

• Arturo Magidin, "What do elements in a localized quotient ring look like?", Mathematics Stack Exchange.
• Atiyah, M. F., & Macdonald, I. G. (1969). Introduction to commutative algebra. Addison-Wesley.
• Bourbaki, N. (1972). Commutative algebra. Hermann.

Further Reading

• Eisenbud, D. (1995). Commutative algebra. Springer-Verlag.
• Lang, S. (1996). Algebra. Springer-Verlag.
• Zariski, O., & Samuel, P. (1958). Commutative algebra. Van Nostrand.
  

Introduction

In our previous article, we explored the concept of "both sides collapse" and the correct map for a past MSE post concerning localization of quotient rings. This phenomenon is a crucial tool in understanding the properties of rings and their quotient rings. In this Q&A article, we will address some common questions and provide further clarification on this topic.

Q: What is the meaning of "both sides collapse"?

A: The concept of "both sides collapse" refers to a situation where the localization of a quotient ring at a prime ideal $P$ results in a ring that is isomorphic to the localization of the original ring $R$ at $P$. In other words, both sides of the quotient ring collapse to the same ring.

Q: How does the localization of a quotient ring work?

A: The localization of a quotient ring $R/I$ at a prime ideal $P$ is defined as the set of equivalence classes of fractions $\frac{a + I}{s + I}$, where $a \in R$ and $s \in R \setminus P$. The equivalence relation is defined as $\frac{a + I}{s + I} \sim \frac{b + I}{t + I}$ if and only if there exists $u \in R \setminus P$ such that $u(at - bs) = 0$. The operations of addition and multiplication are defined as $\frac{a + I}{s + I} + \frac{b + I}{t + I} = \frac{at + bs}{st}$ and $\frac{a + I}{s + I} \cdot \frac{b + I}{t + I} = \frac{ab}{st}$, respectively.

Q: What is the correct map from the localized quotient ring to the localization of the original ring?

A: The correct map from the localized quotient ring to the localization of the original ring is given by the following formula:

$$\frac{a + I}{s + I} \mapsto \frac{a}{s}$$

where $a \in R$ and $s \in R \setminus P$. This map is well-defined and is an isomorphism if and only if both sides collapse.

Q: Can you provide an example to illustrate the concept of "both sides collapse"?

A: Let's consider an example. Suppose we have a ring $R = \mathbb{Z}$ and a prime ideal $P = (2)$. We can define a quotient ring $R/I = \mathbb{Z}/(2)$, where $I = (2)$. The localization of $R$ at $P$ is given by $R_P = \mathbb{Z}_2$, and the localization of the quotient ring is given by $(R/I)_P = (\mathbb{Z}/(2))_2$. If we consider the map from the localized quotient ring to the localization of the original ring, we get:

$$\frac{1 + (2)}{1 + (2)} \mapsto \frac{1}{1}$$

This map is an isomorphism, and both sides collapse.

Q: What are some common mistakes to avoid when working with localization of quotient rings?

A: Some common mistakes to avoid when working with localization of quotient rings include:

• Not checking if the prime ideal $P$ is contained in the ideal $I$.
• Not verifying that the map from the localized quotient ring to the localization of the original ring is well-defined.
• Not checking if the map from the localized quotient ring to the localization of the original ring is an isomorphism.

Q: How can I apply the concept of "both sides collapse" to my research?

A: The concept of "both sides collapse" can be applied to various areas of research, including:

• Algebraic geometry: The concept of "both sides collapse" can be used to study the properties of algebraic varieties and their quotient rings.
• Commutative algebra: The concept of "both sides collapse" can be used to study the properties of commutative rings and their quotient rings.
• Number theory: The concept of "both sides collapse" can be used to study the properties of number fields and their quotient rings.

Conclusion

In conclusion, the concept of "both sides collapse" and the correct map for a past MSE post concerning localization of quotient rings are essential tools in understanding the properties of rings and their quotient rings. By addressing common questions and providing further clarification, we hope to have provided a deeper understanding of this phenomenon. We encourage readers to explore the concept of "both sides collapse" and its applications in various areas of research.

References

• Arturo Magidin, "What do elements in a localized quotient ring look like?", Mathematics Stack Exchange.
• Atiyah, M. F., & Macdonald, I. G. (1969). Introduction to commutative algebra. Addison-Wesley.
• Bourbaki, N. (1972). Commutative algebra. Hermann.

Further Reading

• Eisenbud, D. (1995). Commutative algebra. Springer-Verlag.
• Lang, S. (1996). Algebra. Springer-Verlag.
• Zariski, O., & Samuel, P. (1958). Commutative algebra. Van Nostrand.