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The following is taken from Algebra Vol 2 by Luthar and Passi

Background

$\textbf{Definition 6.6.}$ The ring $(A-\mathfrak{p})^{-1}A$ is a local ring and is called the $\textit{localization of $A$ at its prime ideal}$ $\mathfrak{p}$ be denoted by $A_\mathfrak{p}$, and it is defined as $$A_\mathfrak{p}=\{a/s\mid a\in A, s\not\in\mathfrak{p}\}$$

If $\mathfrak{a}$ is an ideal of $A$, it is customary to denote the extension of $\mathfrak{a}A_\mathfrak{p}=(A-\mathfrak{p})^{-1}\mathfrak{a}$ of $\mathfrak{a}$ in $A_\mathfrak{p}$ by $\mathfrak{a}_\mathfrak{p}$. Thus $\mathfrak{a}_\mathfrak{p}$ consists of elements of the form $a/s$, $a$ in $\mathfrak{a}$ and $s\not\in \mathfrak{p}$.

$\textbf{2.1.1 Proposition.}$ The ring $A_\mathfrak{p}$ is a local ring with maximal ideal $\mathfrak{p}A_\mathfrak{p}$. As $\mathfrak{q}$ runs once through the prime ideals of $A$, contained in $\mathfrak{p}$, the ideal $\mathfrak{q}A_\mathfrak{p}$ of $A_\mathfrak{p}$ runs once through the prime ideals of $A_\mathfrak{p}$. For any ideal $\mathfrak{a}$ of $A$, contained in $\mathfrak{p}$, we have

$$A_\mathfrak{p}/\mathfrak{a}_\mathfrak{p}\cong (A/\mathfrak{a})_{\mathfrak{p}/\mathfrak{a}}\quad(*)$$.

In particular, $A_\mathfrak{p}/\mathfrak{p}A_\mathfrak{p}$ is isomorphic to the field of fractions of $A/\mathfrak{p}$.

Question

For 2.1.1 Proposition above, would the two sets $A_\mathfrak{p}/\mathfrak{a}_\mathfrak{p},(A/\mathfrak{a})_{\mathfrak{p}/\mathfrak{a}}$ in the isomorphism of the proposition in set builder notation in terms of coset representation should be as follows:

let $m,s\in A$, and $\bar{(\frac{m}{s})}=\frac{m}{s}+\mathfrak{a}_{\mathfrak{p}}$

$\frac{A_\mathfrak{p}}{\mathfrak{a}_\mathfrak{p}}=\{\bar{(\frac{m}{s})}\mid m\in A, s\not\in P\}=\{\frac{m}{s}+\mathfrak{a}_{\mathfrak{p}}\mid m\in A, s\not\in P\}$

and since $(\frac{A}{\mathfrak{a}})_{\frac{\mathfrak{p}}{\mathfrak{a}}}$ denotes the localization of the quotient ring $\frac{A}{\mathfrak{a}}$ at the ideal $\frac{\mathfrak{p}}{\mathfrak{a}}$. Then,

let $\bar{m}, \bar{s}\in \frac{A}{\mathfrak{a}}$ where $\bar{m}=m+\mathfrak{a}, \bar{s}=s+\mathfrak{a}$. So

$(\frac{A}{\mathfrak{a}})_{\frac{\mathfrak{p}}{\mathfrak{a}}}=\{\frac{\bar{m}}{\bar{s}}\mid \bar{s}\not\in \frac{\mathfrak{p}}{\mathfrak{a}}\}=\{\frac{m+\mathfrak{a}}{s+\mathfrak{a}}\mid s+\mathfrak{a} \not\in \frac{\mathfrak{p}}{\mathfrak{a}} \}$

Thank you in advance.

Seth
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  • Also by seeing your previous questions, let me stress how important it is to NOT put much focus on the underlying sets of these rings. This already starts with the quotient ring, forget about cosets. They are rings satisfying certain universal properties, and this is how you can work with them, efficiently and without any pain. I am afraid that it is very unlikely that you can read and understand a book on algebra when you continue your current approach ("everything is a set"). – Martin Brandenburg Feb 17 '25 at 19:40
  • @MartinBrandenburg I keep seeing in the topic of localizations, in proofs of various theorems where there are statements like: instead of doing the proof via complicated calculations, it is better to use the universal properties of mapping. I don't undertand why that is. Also, in some other theorems involving isomorphisms, I see the use of exact sequences. However in youtube lecture videos, one professor states that it is important to think in terms of coming up with explicit homomorphism maps so that you see what is going on.... – Seth Feb 17 '25 at 21:16
  • @MartinBrandenburg ... I mean all three approach end up at the same destination of resolving the question. In the end, isn't it a matter of personal perference? – Seth Feb 17 '25 at 21:17
  • @Seth: Often, it obscures what is actually happening. As you go into more detail, let's say, algebraic geometry, then you will have to see the underlying geometric picture, and for stuff like that, having to write down everything is tedious and irrelevant. At the elementary level, understanding what localisation or quotienting is matters way more than what they look as sets. And later on, looking at sets and explicit descriptions adds to the difficulty. – ultralegend5385 Feb 18 '25 at 10:24
  • @Seth: (contd.) And while starting in algebra, it is surely worth it to see both the sides, but understanding why $A_{\mathfrak a}/\mathfrak{p}{\mathfrak a}\simeq(A/\mathfrak{a}){\mathfrak{p}/\mathfrak{a}}$ holds is much more important than seeing as them as sets and noticing they look somewhat same. – ultralegend5385 Feb 18 '25 at 10:27
  • @ultralegend5385 thank you for the comments. I am still confuse about why Martin Brandenburg means when he says "...it is to NOT put much focus on the underlying sets of these rings. This already starts with the quotient ring, forget about cosets. They are rings satisfying certain universal properties, and this is how you can work with them". There are two things I am confused about, (1) what he means by consider them as quotient rings as rings and forget about the cosets. (2) the whole emphasis on universal mapping properties... – Seth Feb 18 '25 at 16:03
  • @ultralegend5385 ...for (1) when encountering a new topic, don't people work hard to try to understand precisely what a notation says and how is expressed down to set builder notations, this way, they can read the proofs of various theorems in the topic easier? I thought that is what everyone who deals with higher mathematics do. I mean if you look at tensor analysis, the entirety of that subject is just how to handle/manipulate those upper and lower indices well when using them to describe the physical situations in science and engineering. For (2) I understand.... – Seth Feb 18 '25 at 16:07
  • @ultralegend5385 ...that the point of universal mapping properties in the notion of category theory is for between two objects, one can to find the best isomorphism amongst all possible isomorphisms. But that is an exercise after proving the isomorphism between two said objects, then one goes ahead with the with doing the universal mapping property exercise. Also, the proving of isomorphism and the satisfying of universal mapping property are two separate and different tasks. I could be wrong since I never have to use universal mapping property of a mathematical object to solve anything... – Seth Feb 18 '25 at 16:12
  • @ultralegend5385 ...other than dealing directly with questions in category theory. It seems like universal mapping property is some sort of religious philosophy in the mathematical sense. I would surely like to hear your explanations on the two issues I have when it comes to what Martin Brandenburg has stated. – Seth Feb 18 '25 at 16:14
  • @Seth: (2) Universal properties are a category-theoretic notion, but there's more to them. It is not a religious philosophy, but the point is, it allows you to be more general in some sense. Why would you want to prove a similar result differently for topological spaces, rings, groups, modules, etc. using their set structure, when you can, once and for all, settle the proof for all of them by understanding the universality of the objects at hand? (...) – ultralegend5385 Feb 20 '25 at 12:10
  • (contd.) Again, as I said earlier, it is important to see what is happening with set theory when you begin, but you should understand both sides, and you will find that universalities are easier to understand and better to use. For (1), that is not always exactly what people do in higher mathematics. They surely try their best to understand what a notation means, but more about what the notion means. Again, it is very much dependent on the specific subject at hand, but I can guarantee that understanding the notions is much more of an important task than just understanding the "sets". (...) – ultralegend5385 Feb 20 '25 at 12:14
  • (contd.) As I said, I believe that a beginner should definitely look at the underlying sets, and understand what kind of elements belong to what set. But there should be a balance and not over-emphasis, and one should start understanding the more general ideas past a point when one achieves a reasonable amount of comfortability with the concepts. Looking at sets, after a point I feel, doesn't tell you what's really going on. That doesn't have to be explainable by some category theory diagram magic, but there's more to it than sets looking same. – ultralegend5385 Feb 20 '25 at 12:17

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