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Questions tagged [localization]

For questions regarding the process, consequences, and stability of localizing algebraic structures such as rings, categories, and modules. Not for use with local topological spaces.

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Show that $\equiv$ is a congruence on $M\times S$

I'm sorry if a similar question has been posted before, but I was unable to find one based on my searches. This is an extra practice problem for a number theory class. I've been trying to prove this ...
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1answer
22 views

If $\mathfrak{p}$ is a prime such that $M_\mathfrak{p} \neq 0$, then $\mathfrak{p}$ contains an associated prime of $M$

I am studying from Serge Lang's Algebra (3rd edition), and in Chapter X Noetherian Rings and Modules, $\S2$ Associated Primes, we have the following proposition: Proposition 2.10. Let $A$ be ...
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1answer
25 views

Localization of $k[t]$ at a prime ideal is not a finitely generated $k[t]$-algebra?

Let $k$ be a field (infinite and algebraicaly closed), consider the prime ideal $(t)$ in $k[t]$. Consider the localization of $k[t]$ at $(t)$: $k[t]_{(t)}$. Now $k[t]_{(t)}$ is a $k[t]$-algebra. I ...
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1answer
18 views

Commutative ring with finitely many minimal primes [duplicate]

$A$ be a commutative ring with $1$ with finitely many minimal primes $\{p_1,\ldots,p_n\}.$ Then how can I show that $S^{-1}A \cong A_{p_1} \times\cdots \times A_{p_n},$ where $S=A \setminus \bigcup_{i=...
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1answer
55 views

Intersection of the kernels of localization maps

Let $M$ be a finitely generated module over a Noetherian ring $R$. I need to show that for a multiplicately closed subset $U\subset R$, $$\bigcap_{P\in \operatorname{Ass}(M)\\ P\cap U=\emptyset}\ker(M\...
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2answers
28 views

if $M \otimes_A (A_m/mA_m)=0$ for every maximal ideal $m \subset A$, then $M=0$, $M$ finitely generated

Suppose $M$ is a finitely generated $A$-module. Prove that if $M \otimes_A (A_m/mA_m)=0$ for every maximal ideal $m \subset A$, then $M=0$. Subscrpit $_m$ means localization at $m$. First consider ...
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1answer
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The localization of the ring $\mathbb{Z} \times \mathbb{Z}$ at every prime ideal is an integral domain

I want to show that the localization of the ring $\mathbb{Z} \times \mathbb{Z}$ at every prime ideal is an integral domain. $\mathbb{Z} \times \mathbb{Z}$ is not an integral domain since $(0,1)\cdot(...
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1answer
32 views

The map $M[f^{-1}]\to M[f^{-1}g^{-1}]$

Let $M$ be a module over $R$. When one talks about the map $M[f^{-1}]\to M[f^{-1}g^{-1}]$, which map do they mean? My guess is that $M[f^{-1}][g^{-1}] \simeq M[f^{-1}g^{-1}]$, and then the above map ...
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Localization of hereditary rings

A (left) hereditary ring $R$ is a ring for which all submodules of projective (left) $R$-modules are again projective. If $R$ is commutative, and $S\subseteq R$, then $S^{-1}R$ is hereditary :https://...
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$R_{\mathfrak{p}}$ is integrally closed in its total quotient ring, then $R$ is integrally closed in its total quotient ring.

It is known that if $R$ is a Noetherian ring, then an element $x\in K(R)$ in its total quotient ring belongs to $R$ iff the image of $x$ in $K(R)_{\mathfrak{p}}$ belongs to $R_{\mathfrak{p}}$ for ...
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30 views

On modules in which tensor of each two elements is commutative

Let $R$ be a commutative ring with unity. Let $M $ be a finitely generated $R$-module. Is it true that the following conditions are equivalent : (1) In $M \otimes_RM$, we have $m\otimes n=n\otimes m,...
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1answer
18 views

An equivalent condition for the localization map $M\to M_P$ to be injective

How do I show that given a prime ideal $P\subset R$, the map of $M\to M_P, m\mapsto m/1$ is injective if and only if $R\setminus P$ contains no elements killing a nonzero element of $M$? One ...
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3answers
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Show that every non-zero ideal of the ring of rationals with odd denominator is generated by $2^{n}$ [duplicate]

Let $R \subset \mathbb{Q}$ be the subring $\left\{\frac{a}{b} \mid a, b \in \mathbb{Z}, b \text { odd }\right\}$. Prove that the ideals of $R$ are the zero ideal $\{0\}$ and $2^{n} R$ for $n \geq 0$. ...
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2answers
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$K[X]_X$ is not integral over $K[X]$

The localization $K[X]_{X}$ is a ring extension of $K[X].$ I want to show that $K[X]_X$ is not integral over $K[X]$ using lying above. I tried to find a maximal ideal in $K[X]_X$ whose contraction in ...
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35 views

Is it possible to localise the renormalisation group?

The renormalisation group allows you to consider what happens to a model when considered on different spatial scales. It is not a group, reflecting the fact that you can always move to a larger ...
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1answer
24 views

Do the maps of rings coincide under the condition of reducedness and localization assumption?

Say $f,g:B\rightarrow A$ are maps of rings such that $A$ is reduced and $i_{\mathfrak p}\circ f=i_{\mathfrak p}\circ g $ for all primes $\mathfrak p\subset A$ where $i_{\mathfrak p}:A\rightarrow A_{\...
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1answer
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Localization of the polynomial ring at a prime ideal modulo maximal ideal is isomorphic to polynomial ring modulo prime ideal.

Let $p \in K[T]$ irreducible, s.t. $\text{LC}(p) = 1$. Then $$ K[T]/(p) \cong K[T]_{(p)}/pK[T]_{(p)}.$$ What I have is: \begin{align*} &K[T] \hookrightarrow K[T]_{(p)} \text{ and } K[T]_{(p)} \...
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Prove Valuation Ring must be a ring localized at a prime

Let $A$ be a principal entire ring, and let $K$ be its quotient field. Let $\mathfrak{o}$ be a valuation ring of $K$ containing $A$. Show that $\mathfrak{o} = A_{(p)}$ for some prime element $p$. [...
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1answer
54 views

Do we have $M=A$?

Let $A$ be a commutative ring with unity and $M$ an ideal of $A$. If $\forall\mathfrak q\in \operatorname{Spec}A$, the canonical morphism $M_{\mathfrak q}\to A_{\mathfrak q}$ is an isomorphism, do we ...
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1answer
49 views

Ring of fractions of $\mathbb Z_6$

I am studying commutative algebra and I have a problem in calculating the fraction rings. For example, it would be a great help someone help me to calculate the ring of fraction if $ S= \{1,3,5\} $ ...
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1answer
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Is $\mathbb{C}[[t]]$ isomorphic to the localization of $\mathbb{C}[t]$ at the maximal ideal $(t)$?

It seems to me that the power series ring $\mathbb{C}[[t]]$ is isomorphic to the localization of $\mathbb{C}[t]$ at the maximal ideal $(t)$, but I am not sure.
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1answer
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Infinite direct sum of p-adic integers is not p-adic

Studying Bousfield localization I stumbled upon this elementary example: we start with $\mathcal{D}$ the derived category of $p$-local abelian groups and we can consider the Bousfield class of $\...
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1answer
92 views

Finding the field of fractions of a quotient of a polynomial ring?

This should be very basic but I am having a bit of trouble finding the field of fractions for quotients of polynomial rings over a field. The specific example I am having trouble with is the following:...
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1answer
37 views

The localization of an ideal is equal to the localization of the ring

Suppose $m\subset R$ is a maximal ideal. Suppose $I\subset R$ is an ideal. I'm trying to understand these claims: If $m$ does not contain $I$, then $I_m=R_m$ as localizations of $R$-modules. If $m$ ...
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1answer
60 views

Equality of fractions in localization as quotient

Let $S\subset A$ be a subset of a commutative ring. Consider the polynomial algebra $A[x_s\mid s\in S]$ and define the localization $A_S$ as the quotient by the ideal $ (sx_s-1\mid s\in S)\...
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0answers
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There exists an element $x$ of $M$ such that $W^{-1}R\dfrac{x}{1}=W^{-1}Rz$ and $\operatorname{ann}_R x=\operatorname{ann}_{W^{-1}R} z\cap R$

Let $R$ be a Noetherian ring, $W$ a multiplicative subset of $R$, $M$ an $R$-module, and $z$ an element of $W^{-1}M$. Show that there exists an element $x$ of $M$ so that $W^{-1}R\dfrac{x}{1}=W^{-1}Rz$...
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1answer
48 views

In what condition, $S^{-1}I = S^{-1}R$ if and only if $S \cap I \neq \emptyset$

$R$ is a commutative ring. $S^{-1}R$ is a ring of quotients and $S^{-1}I$ is an extension of ideal $I$ in $S^{-1}R$. In Algebra by Hungerford, Theorem III.4.8, page 146, he says when $R$ is a ...
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1answer
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Which primes intersect the submonoid $f^\mathbb{N}(1+fA)\subset A$?

Let $A$ be a commutative ring and $f\in A,I\vartriangleleft A$. We have obvious submonoids $f^\mathbb{N},1+I\subset A$. The submonoid $f^\mathbb{N}\subset A$ is saturated, and its complement is the ...
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1answer
42 views

Can two different multiplicative systems give same localisation?

Can we have two different multiplicative systems $S_1$ and $S_2$ in $\mathbb{Z}$ having same localisations $\mathbb{Z}_{S_1}=\mathbb{Z}_{S_2}$? I have a trivial solution by taking two different ...
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1answer
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Localisation of $\mathbb{Z}_{12}$ by the multiplicative set $T=\{2^k|k\geq 0\}$ [closed]

What are the elements of the localisation of $\mathbb{Z}_{12}$ with respect to the multiplicative set $T=\{2^k(\bmod 12)\mid k\geq 0\}$?
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1answer
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Localization correspondence

This is taken from Neukirch's algebraic number theory Proposition 12.3. Proposition (12.3). If $a\neq 0$ is an ideal of an order (one dimensional Noetherian integral domain) $o$, then: $o/a = \...
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2answers
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Proving $R-S$ contains a prime ideal when $S$ is a multiplicative set

I'm mainly trying to prove that If $0\not \in S\subseteq R$ is a multiplicative subset of a commutative ring $R$ with identity. Then $R-S$ contains a prime ideal. Now, by using Zorn's lemma, ...
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1answer
74 views

Does $\frac{b}{s} \in S^{-1} I$ directly implies that $b \in I$?

Let $R$ be a commutative ring with identity $1_R$, $0 \not \in S\subseteq R$ be a multiplicative set, and $I\subseteq R$ be an ideal of $R$.Consider the ring of quotients $S^{-1}I$. I was trying to ...
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3answers
57 views

Prove $R_{\mathfrak p}$ has only one maximal ideal $\mathfrak pR_{\mathfrak p}$

$\mathfrak p$ is prime ideal of commtative ring $R$. localization $R_{\mathfrak p}:=(R-{\mathfrak p})^{-1}R$. We know $\mathfrak pR_{\mathfrak p}=(R-{\mathfrak p})^{-1}\mathfrak p$ is prime ideal of $...
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0answers
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$R_p$ is a field iff for any $x \in p$ there exists $y \not\in p$ such that $xy = 0$

Show that the localization at p, prime ideal, $R_p$ is a field iff for any $x \in p$ there exists $y \not\in p$ such that $xy = 0$ I know there is a similar question where R is an Noetherian ring, ...
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0answers
53 views

Question regarding localization of polynomial ring

I made an exercise that went as follows: Suppose $R = \frac{\mathbb{R}[x,y]}{(xy)}$. Define the multiplicative set $$ S = \left\{ 1 + (xy), x + (xy), x^2 + (xy), \ldots \right\}.$$ In the exercise, I ...
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1answer
47 views

What does it mean for localization to be same if multiplicative set is different? $T^{-1}R = S^{-1}R$?

If R is a ring and p is a prime ideal, I was told that somehow $Frac(R/p)$ was the localisation $(R \setminus p)^{-1}R/p$. I thought it was more like the localisation $(R/p \setminus\{p\})^{-1}R/p$ ...
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35 views

Integrally closed domains, where is the reduced hypothesis used?

The question is to show, given a local domain $A$, with maximal ideal $(\pi)$, $B$ a domain containing $A$, $S=A-(\pi)$. If $S^{-1}B$ is integrally closed, and $B/\pi B$ reduced, then $B$ is ...
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1answer
94 views

When is the localization map injective?

Suppose that $R$ is a commutative ring and $S\subset R$ is multiplicatively closed subset, i.e. $1\in S$ and if $a,b\in S$ then $ab\in S$. Consider the natural mapping $\phi:R\to S^{-1}R$ defined by $\...
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1answer
43 views

Question about the ring $A_P$ of regular functions on a variety $V$

Note: I am doing an introductory course in Commutative Algebra and not Algebraic Geometry; This is just a quick application I am trying to understand. Also, rings are commutative with $1$. When ...
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0answers
58 views

localization of a polynomial ring at a prime ideal

My question is about the hint of exercise 5.1 of Matsumura: Let $k$ be a field,and $R=k[X_{1},\dots,X_{n}]$ and let $\mathfrak{p}\in \operatorname{Spec} R$. Set $k[X_{1},\dots,X_{n}]/\mathfrak{p}=k[...
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2answers
60 views

Let $R \subseteq S$ be two local PIDs with the same field of fractions, then $R=S$.

Let $R$ and $S$ be two local principal ideal domains with the same field of fractions $K$. I want to show that if $R\subseteq S$ then $R=S$. I will denote as $\mathfrak{m}_R=(m_R)$ and $\mathfrak{m}...
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1answer
32 views

Valuation ring and localizations [closed]

Theorem. Let $D$ be an integral domain with identity. The following conditions are equivalent. (1) $D_P$ is a valuation ring for each proper prime $P$ in $D$. (2) $D_M$ is a valuation ring for each ...
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1answer
35 views

Localisation of $\mathbb{Z}/(p^k)$.

I was looking at the wiki that explains localization. It says that the only way to localize $\mathbb{Z}/(p^k)$ is $\{0\}$. The argument is that the elements of $\mathbb{Z}/(p^k)$ are either units or ...
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0answers
68 views

When does localisation behave badly?

Localisation seems to be a very useful tool in commutative algebra/number theory, and it seems like in every case I've come across, it behaves incredibly well. By behaves well, I mean that it is ...
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0answers
47 views

Localization of finitely generated algebra

Let $R$ be a reduced finitely generated algebra over $\Bbb Z$. Let $T$ be a finite set of prime ideals of $R$. Let $S = \bigcap_{p \not\in T} R \setminus p$. 1) Is it true that $A := S^{-1}R$ is ...
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0answers
21 views

Find maximal ideals in semi local ring with singular

I am trying to analyze the normalization $N$ of a local ring $A_{\mathfrak{m}}$ of a variety with Singular. The normalization (integral closure in its total ring of fractions) is semi-local with its ...
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1answer
72 views

Isomorphism between localizations of graded ring $S_{(P)} \cong [S_{(f)}]_{PS_f \cap S_{(f)}}$

I know that if $S$ is a graded ring, and $f$ is a homogeneous element of positive degree, then there is a bijection between the homogeneous prime ideals of the localization $S_f$ and the prime ideals ...
3
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1answer
49 views

When does it suffice to show statements about rings only for the local ring after localizing at a prime?

I'm learning commutative Algebra with the book from Eisenbud. But I'm having trouble understanding some of his Proofs. Often we have a Statement about a ring R or an R-module M, which we'd like to ...
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0answers
39 views

localization at one element, regular element

I'm trying to prove that An element $\frac{a}{f^n}$ of $A_f$ is regular iff $\frac{a}{f^n}$ is regular in $A_p$, for all prime ideals $p$ of $A$ such that $f\notin p$. Where $A$ is a unitary ...