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 of quotient rings is a crucial topic of study. Localization is a process that allows us to create a new ring from an existing one by inverting certain elements. However, this process can lead to some interesting and counterintuitive results, such as the phenomenon of "both sides collapse." In this article, we will delve into the meaning of "both sides collapse" and explore the correct map for a past MSE post concerning the localization of quotient rings.

Background

According to the answer provided by Arturo Magidin for the post "What do elements in a localized quotient ring look like?", we are given the following information:

Given a ring $R$ and a prime ideal $P$, then we can define the localization of $R$ at $P$, denoted by $R_P$, as the set of equivalence classes of pairs $(r, s)$, where $r \in R$ and $s \in R \setminus P$, under the equivalence relation $(r, s) \sim (r', s')$ if and only if there exists $t \in R \setminus P$ such that $t(rs' - r's) = 0$.

This definition allows us to create a new ring $R_P$ from the original ring $R$ by inverting the elements outside the prime ideal $P$. However, this process can lead to some unexpected consequences, such as the phenomenon of "both sides collapse."

What is "Both Sides Collapse"?

So, what exactly is "both sides collapse"? In the context of localization of quotient rings, "both sides collapse" refers to the phenomenon 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 the same prime ideal $P$. In other words, both sides of the quotient ring collapse to the same ring.

To understand this phenomenon, let's consider an example. Suppose we have a ring $R = \mathbb{Z}$ and a prime ideal $P = (2)$. We can define the localization of $R$ at $P$ as $R_P = \mathbb{Z}_2$. Now, suppose we take the quotient ring $R/P = \mathbb{Z}/2\mathbb{Z}$. We can then define the localization of this quotient ring at the same prime ideal $P$ as $(R/P)_P = (\mathbb{Z}/2\mathbb{Z})_2$.

The Correct Map

So, what is the correct map for the localization of the quotient ring $(R/P)_P$? To answer this question, we need to understand the relationship between the localization of a ring and the localization of its quotient ring.

Let's consider the following commutative diagram:

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

In this diagram, the top row represents the localization of the ring $R$ at the prime ideal $P$, while the bottom row represents the localization of the quotient ring $R/P$ at the same prime ideal $P$. The vertical arrows represent the natural projections from the original ring to its localization and from the quotient ring to its localization.

The Key Insight

The key insight here is that the localization of a quotient ring at a prime ideal $P$ is isomorphic to the localization of the original ring $R$ at the same prime ideal $P$. In other words, both sides of the quotient ring collapse to the same ring.

To see this, let's consider the following map:

$$\phi: (R/P)_P \rightarrow R_P$$

This map is defined as follows: given an element $(\bar{r}, \bar{s}) \in (R/P)_P$, where $\bar{r} \in R/P$ and $\bar{s} \in R/P \setminus P$, we can lift this element to an element $(r, s) \in R_P$, where $r \in R$ and $s \in R \setminus P$. The map $\phi$ is then defined as $\phi((\bar{r}, \bar{s})) = (r, s)$.

The Proof

To prove that the map $\phi$ is an isomorphism, we need to show that it is a bijective homomorphism.

First, let's show that $\phi$ is a homomorphism. Given two elements $(\bar{r}, \bar{s})$ and $(\bar{r'}, \bar{s'}) \in (R/P)_P$, we can lift these elements to elements $(r, s)$ and $(r', s') \in R_P$. Then, we have:

$$\phi((\bar{r}, \bar{s}) + (\bar{r'}, \bar{s'})) = \phi((\bar{r} + \bar{r'}, \bar{s} \bar{s'})) = (r + r', ss')$$

$$\phi((\bar{r}, \bar{s})) + \phi((\bar{r'}, \bar{s'})) = (r, s) + (r', s') = (r + r', ss')$$

Therefore, $\phi$ is a homomorphism.

Next, let's show that $\phi$ is bijective. To show that $\phi$ is injective, suppose that $\phi((\bar{r}, \bar{s})) = \phi((\bar{r'}, \bar{s'}))$. Then, we have:

$$(r, s) = (r', s')$$

This implies that $\bar{r} = \bar{r'}$ and $\bar{s} = \bar{s'}$. Therefore, $(\bar{r}, \bar{s}) = (\bar{r'}, \bar{s'})$, and $\phi$ is injective.

To show that $\phi$ is surjective, suppose that $(r, s) \in R_P$. Then, we can define an element $(\bar{r}, \bar{s}) \in (R/P)_P$ as follows:

$$\bar{r} = \bar{r} \in R/P$$

$$\bar{s} = \bar{s} \in R/P \setminus P$$

We have:

$$\phi((\bar{r}, \bar{s})) = (r, s)$$

Therefore, $\phi$ is surjective.

Conclusion

In conclusion, the localization of a quotient ring at a prime ideal $P$ is isomorphic to the localization of the original ring $R$ at the same prime ideal $P$. This phenomenon is known as "both sides collapse." The correct map for the localization of the quotient ring is given by the natural projection from the quotient ring to its localization.

References

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

Further Reading

• Eisenbud, D. (1995). Commutative algebra with a view toward algebraic geometry. Springer-Verlag.
• Lang, S. (1996). Algebra. Springer-Verlag.
• Zariski, O., & Samuel, P. (1958). Commutative algebra. Springer-Verlag.
  Q&A: Understanding the Concept of "Both Sides Collapse" in Localization of Quotient Rings =====================================================================================

Introduction

In our previous article, we explored the concept of "both sides collapse" in the context of localization of quotient rings. This phenomenon occurs when the localization of a quotient ring at a prime ideal $P$ is isomorphic to the localization of the original ring $R$ at the same prime ideal $P$. In this article, we will answer some frequently asked questions about this concept.

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

A: The significance of "both sides collapse" lies in its implications for the structure of the localized quotient ring. When both sides collapse, the localized quotient ring is isomorphic to the localized original ring, which means that the quotient ring does not introduce any new structure that is not already present in the original ring.

Q: How does "both sides collapse" relate to the localization process?

A: The localization process involves inverting certain elements in the original ring to create a new ring. When both sides collapse, the localization of the quotient ring at a prime ideal $P$ is isomorphic to the localization of the original ring $R$ at the same prime ideal $P$. This means that the localization process does not change the underlying structure of the ring, but rather reveals the existing structure in a new way.

Q: Can "both sides collapse" occur in any ring?

A: No, "both sides collapse" can only occur in rings that have a certain property called "localization-friendliness". This property means that the ring has a well-behaved localization process, and that the localization of the quotient ring at a prime ideal $P$ is isomorphic to the localization of the original ring $R$ at the same prime ideal $P$.

Q: What are some examples of rings that exhibit "both sides collapse"?

A: Some examples of rings that exhibit "both sides collapse" include:

• The ring of integers $\mathbb{Z}$ localized at the prime ideal $(2)$
• The ring of polynomials $k[x]$ localized at the prime ideal $(x)$
• The ring of matrices $M_n(k)$ localized at the prime ideal $(I_n)$

Q: How can I determine if a ring exhibits "both sides collapse"?

A: To determine if a ring exhibits "both sides collapse", you can use the following steps:

1. Localize the ring at a prime ideal $P$ to create a new ring.
2. Localize the quotient ring at the same prime ideal $P$ to create another new ring.
3. Check if the two new rings are isomorphic.

If the two new rings are isomorphic, then the ring exhibits "both sides collapse".

Q: What are some applications of "both sides collapse"?

A: "Both sides collapse" has several applications in algebra and geometry, including:

• The study of singularities in algebraic geometry
• The study of local cohomology in commutative algebra
• The study of representation theory in algebra

Conclusion

In conclusion, "both sides collapse" is a fascinating phenomenon that occurs in the context of localization of quotient rings. It has significant implications for the structure of the localized quotient ring and has several applications in algebra and geometry. We hope that this Q&A article has provided a helpful introduction to this concept and has sparked further interest in the study of localization of quotient rings.

References

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

Further Reading

• Eisenbud, D. (1995). Commutative algebra with a view toward algebraic geometry. Springer-Verlag.
• Lang, S. (1996). Algebra. Springer-Verlag.
• Zariski, O., & Samuel, P. (1958). Commutative algebra. Springer-Verlag.