Questions tagged [ring-isomorphism]

This tag is for questions regarding to Ring Isomorphisms, a ring homomorphism having a $2$-sided inverse that is also a ring homomorphism. Isomorphic rings have all their ring-theoretic properties identical. One such ring can be regarded as "the same" as the other.

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If ring endomorphisms of finite dimensional vector spaces are isomorphic => Vector spaces and division rings are isomorphic

Let $D$, $E$ be division rings and $_ D M$ and $_ E N$ finite-dimensional vector spaces. Show that $End_D(M) \cong End_E(N)$, if and only if $D \overset{\psi}{\cong} E$ and $\dim_D(M) = \dim_E(N)$. ($\...
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Isomorphism between $\mathbb{Z}[C_n]$ and $\mathbb{Z}[X]/(X^n-1)$ [duplicate]

I'm trying to prove that $\mathbb{Z}[C_n] \cong \mathbb{Z}[X]/(X^n-1)$ where $C_n$ is the cyclic group of order $n$. Let $t$ be a generator of $C_n$. There is a unique ring homomorphism $$\Phi:\mathbb{...
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Isomorphism in the quotient ring of a localization

Let $R$ be a ring with unity and suppose $I$ is a maximal ideal of $R$ such that $M = R \setminus I$ is a right denominator set for $R$. Is it true that $R_M/I_M \simeq R/I$, where $R_M$ is the right ...
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Subset of $\operatorname{End}_A(M)$ faithful implies $M$ is faithful.

Let $A$ be a complete intersection ring, $M$ be a finite $A$-module of positive depth (over the maximal ideal), and $B$ be the image of $A$ in $\operatorname{End}_A(M)$. It is easy to show that $B$ ...
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1 vote
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Are field isomorphisms necessarily linear isomorphisms?

Let’s say we have a finite field extension $L/K$. Suppose we have that $L(\alpha )$ and $L(\beta )$ are isomorphic as fields (finite extensions) with an isomorphism $\phi $. Now viewing $L(\alpha ) $ ...
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2 answers
38 views

What is the example of non canonical isom? [closed]

In mathematics, isomorphism $f$ between two objects are sometimes called canonical when $f$ is unique as a map, in other words, everyone choose the same map, no different map gives the isomorphism. ...
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1 answer
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Are two matrices isomorphic? (as rings and as group) [closed]

Assume that $M_2(R) , M_3(R)$ are matrices with real cells $2 \times 2$ , $3 \times 3$ respectively. 1)Are $M_2(R) , M_3(R)$ Isomorphic as rings under addition and multiplication ? why? 2) Are $M_2(R) ...
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2 answers
71 views

The ring $\frac{\mathbb{Z}[x]}{\langle x^n \rangle}$ is isomorphic to?

It is an easy exercise to show that $\frac{\mathbb{Z}[z]}{\langle x \rangle }$ is isomorphic to $\mathbb{Z}$ using the fundamental theorem of ring homomorphisms. I was wondering if the quotient ring $\...
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Why $(x) \cap (1 - x) = (x^2 - x)\,$ [in proof of $\,\Bbb C[x]/(x^2 - x) \cong \Bbb C\times \Bbb C$ by the Chinese Remainder Theorem] [duplicate]

I am trying to solve problem 4B from Napkin: Show that $\mathbb{C}[x]/(x^2 - x) \cong \mathbb{C} \times \mathbb{C}$. I was told that this is possible to prove using the generalized Chinese Remainder ...
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1 vote
1 answer
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Showing that End$_{FG}V$ is isomorphic to $F$ as a ring.

Let $G$ be a finite group, $F$ be an algebraically closed field, and $V$ an irreducible $FG$-module. I want to show that End$_{FG}V$ is isomorphic to $F$ as a ring. I managed to get that End$_{FG}V$ ...
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Ring isomorphism between $(k[x,y]/(xy))/(x+y-a)$ and $k\times k$

Let $k$ be a field and let $a\in k$ be a non zero element. Consider the quotient ring $A = k[x,y]/(xy)$. If $f\in k[x,y]$ we denote by $\overline{f}$ its image in $A$. Now consider the quotient ring $...
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2 answers
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Isomorphism of quotients implies isomorphism of the denominator [closed]

Let $A$ be a commutative ring and $I \subset J$ two ideals of $A$. Let's suppose we have $A/I \simeq A/J$ as $A$-modules. Using the second isomorphism theorem can we deduce that $I \simeq J$ ? I ...
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Proving an isomorphism difficulty [duplicate]

Let $\phi$ be a mapping from R to R, where R is a finite field. My teacher said that to show $\phi$ is an isomorphism, it is enough to show that $\phi$ is injective (one-one) along with being a ...
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Isomorphism between Rings using Chinese Remainder Theorem

Let $A_n(K)$ be the vector space of applications betwen $\mathbb{Z}_n$ and the field $K$, consider it as a ring with the convolution product $(f*g)(t) = \sum_{m \in \mathbb{Z}_n} f(t)g(t-m)$. If $K$ ...
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1 vote
1 answer
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Poor classification of ring morphisms $\mathbb{Z} \to \mathbb{Z}_n$ [duplicate]

I've found issues with questions in my algebra textbook before, and I believe that I've found another erroneous question. L. E. Sigler Algebra Chapter 2 "Rings: Basic Theory" Section 7 "...
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3 votes
1 answer
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Quotient Ring of Gaussian Integers - $\mathbb{Z}[i]/I_n$ is a field iff $q = n^2 + 1$ is prime.

I am working on a problem in which I need to prove that $\mathbb{Z}[i]/I_n$ is a field if, and only if, $q = n^2 + 1$ is prime. Here, $I_n = _{\mathbb{Z}[i]}\langle i -n \rangle$ is a given ideal of ...
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2 votes
2 answers
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Basic Question about Uniqueness of the Real Numbers

I'm trying to write a proof that $\mathbb{R}$ is isomorphic to any ordered field which has the least upper bound property. I'm glancing at Spivak's Calculus, Chapter 30 when I have issues. I am ...
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1 vote
1 answer
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Constructing an isomorphism between two rings

Consider a commutative ring $R$ with unity. How does one show that $R$ is isomorphic to the quotient of a polynomial ring $\mathbb{Z}[x_1,x_2,\dots]$ (possibly infinitely many variables)? I have been ...
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1 answer
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Isomorphism from $\mathbb{Z}[\sqrt{d}]$ to a subring of $M_2(\mathbb{Z})$ [duplicate]

If $d\in \mathbb{Z}$ is square free (we also consider $d=-1$ to be square free), show that the quadratic ring $\mathbb{Z}[\sqrt{d}]$ is isomorphic to the subring $B \subseteq M_2(\mathbb{Z})$ where $$...
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1 vote
2 answers
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Construction of $\mathfrak{J}$

I have what could be taken as a quite naive way of understanding the complex field. I simply define the ring isomorphism: \begin{equation} \phi:\mathbb{R}^2\longmapsto\mathfrak{J} \end{equation} Where ...
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0 answers
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Condition for conormal module of commutative, Noetherian, local ring to have finite length

Let $A$ be a commutative, Noetherian, local ring, $O$ a discrete valuation ring and $\lambda : A \rightarrow O$ be an epimorphism. Let $p=\ker(\lambda)$, and consider the conormal $A$-module $p/p^2$. ...
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2 votes
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For which rings $R$ is $\operatorname{End}_{\mathbb Z}(R)$ isomorphic to $R$ (that is, the only endomorpisms are the multiplication maps)?

Let $(R,+,\cdot)$ be a commutative ring. It is a well-known fact that the ring of $R$-module endomorphisms of $R$, that is $(\operatorname{End}_R(R),+,\circ)$, is isomorphic to $(R,+,\cdot)$. Clearly, ...
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Question about showing that $\mathbb{K}[X,Y]/(XY-1)$ is isomorphic to $\mathbb{K}[T,T^{-1}]$. [duplicate]

I want to show that $\mathbb{K}[X,Y]/(XY-1) \cong \mathbb{K}[T,T^{-1}]$ and I’ve asked myself it is possible to show this by using the euclidean division of a polynomial in $\mathbb{K}[X,Y]/(XY-1)$?
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3 votes
1 answer
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Two Properties of Perfect Rings

I came across Kunz's theorem about the characterization of regular rings in characteristic $p$. In the paper that I am reading, the author uses perfect rings to prove this result. Perfect rings $R$ ...
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-1 votes
1 answer
84 views

Defining isomorphism of rings of matrix [closed]

I'm having some difficulty with the following problem: Define an isomorphism of rings $(\mathbb{Z}/5\mathbb{Z})[X]/(X^2) \cong A$ where $A = \Bigg\{ \begin{pmatrix} a & a-b\\ a-b & a \...
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4 votes
1 answer
144 views

Is this set a ring?

I am studying Ring Theory for the first time in my life- so the following question may be a very silly one. While trying to solve an (unrelated) exercise, this question clicked in me, and it has been ...
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Every isomorphism of commutative ring with 1 with product of two rings is defined by idempotent element $x$ and element $1-x$. [duplicate]

I proved that $f: R \rightarrow P_x \times P_{x\prime}$ where $x$ is impodent element in $R$, $x\prime=(1-x)$ (also impodent) and $P_x=\{rx:r \in R\}$. But I have problem with proving that every ...
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2 votes
2 answers
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How to come up with this isomorphism?

A question from Herstein's Abstract Algebra book goes- Let $(R,+,\cdot)$ be a ring with unit element. Using its elements we define a ring $(\tilde R,\oplus,\odot)$ by defining $a\oplus b = a + b + 1$ ...
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0 votes
4 answers
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Showing that $\mathbb{Z}_{24}$ is not isomorphic to $\mathbb{Z}_{4}\times\mathbb{Z}_6$

In a previous exam assignment, there is a problem that asks for a proof that $\mathbb{Z}_{24}$ and $\mathbb{Z}_{4}\times\mathbb{Z}_6$ are not isomorphic. We have $\mathbb{Z}_{24}$ is isomorphic to $\...
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2 votes
1 answer
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Over a left V-ring, why is every module containing a simple essential submodule is simple?

I've tried to prove the statement $R$ is a $V-$ring if and only if every module containing a simple essential submodule is simple. I know that a ring $R$ is called as a $V-$ring if every simple $R$ ...
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1 vote
0 answers
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Doubts regarding correspondence theorem.

I am having a confusion regarding the correspondence theorem for rings.I want to know if the following is correct. Let $R$ be a ring and $I$ be an ideal of $R$.Consider the sets $\mathcal G$ and $\...
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1 vote
1 answer
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Given a proper field extension $L/K$, can we have $L\cong K$? [duplicate]

Given a proper field extension $L/K$ (that is, $K$ can be considered as a proper sub-field of $L$). Can it still happen that $K\cong L$ via a field-isomorphism? I assume No, but I am utterly ...
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Prove: $F[t]/H[t]$ is isomorphic to $(F/H)[t]$

Let $F$ a commutative and unitary ring, and also $F$ is an integral domain. Let $H$ be a proper ideal of $F$. I want to prove: $F[t]/H[t]$ is isomorphic to $(F/H)[t]$ So my idea is to use the ...
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0 votes
0 answers
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Proof for $P(R) \leq I-rad(R_R)$

The (right) $\textbf{isoradical}$ $I-rad(R_R)$ of a ring $R$ is the intersection of the annihilators of all isosimple right $R-$modules where an $\textbf{isosimple}$ module is defined as a non-zero ...
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0 votes
0 answers
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Completely virtually semisimple modules are direct sums of isosimple modules.

An $\textbf{isosimple}$ module is defined as a non-zero module whose all non­zero submodules are isomorphic to it. An $R$-module $M$ is called $\textbf{virtually semisimple}$ if every submodule of $M$ ...
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1 vote
0 answers
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Some Characterizations of Isoartinian Module

Let $R$ be a ring and $M$ be a right $R$-module. $M$ is called $\textbf{isoartinian}$ if, for every descending chain $M \geq M_1 \geq M_2 \cdots$ of submodules of $M$, there can be found an index $n$ $...
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0 votes
1 answer
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Find all the ideals $\mathfrak{b}$ that fit $\mathfrak{a}\subsetneq\mathfrak{b}\subsetneq\mathbb{R}[X,Y,Z]$,being$\mathfrak{a}=(X^2-2X-4,Y-X^2,Z-X^3)$

Let $\mathbb{K}$ be a field, and $\mathfrak{a}=(X^2-2X-4,Y-X^2,Z-X^3)$ an ideal of $\mathbb{K}[X,Y,Z]$. The first exercise was to see if $\mathfrak{a}$ is prime or maximal in $\mathbb{K}[X,Y,Z]$, ...
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1 answer
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Prove that for a finite field $R$ of characteristic $p$, the mapping $f(a)=a^p$ for every element $a$ of $R$ is an isomorphism.

$R$ is a finite field having characteristic $p(>0)$. Let $f(a)=a^p$ where $a\in R$ be a mapping from $R$ to $R$. I have to show that it is an isomorphism. I am done with the homomorphism part. But ...
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1 vote
0 answers
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How to prove something is NOT a ring isomorphism?

I have a question about rings (I assume ring without the requirement of multiplicative identity element). Suppose, I want to show, for two rings, there is NO ring isomorphism (the rings are not ...
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0 answers
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isomorphism $\mathbb{C} [X,Y]/(X^2 + Y^2 - 1)$

I have to show $\mathbb{C} [X,Y]/(X^2 + Y^2 - 1)$ is isomorphic to $\mathbb{C} [U,V]/(UV - 1)$. First, I would apply the first isomorphism theorem by considering $ \mathbb{C} [X,Y] \rightarrow \mathbb{...
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$\sigma:Z\to K$ is an integral domain. Why is this an explanation for the fact that $\sigma(Z)\cong Z$ or $F_p$?

Someone mentioned Serre's A Course in Arithmetic in a book-recommendation question. I was curious and so I downloaded a copy and read it, but I was already tripped by the opening paragraph in the ...
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8 votes
2 answers
296 views

Prove that $\mathbb Z[i]/(1+3i)\simeq \mathbb Z_{10}$ [duplicate]

I am confused how to do the proof that $\mathbb Z[i]/(1+3i)\simeq \mathbb Z/10\mathbb Z$. It is clear to me intuitively. $1+3i=0$ in the quotient ring, so $3i=-1\implies 9i^2=1\implies 10=0$ in ...
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0 votes
1 answer
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How to prove this isomorphism?

I know about the correspondence between ideals in $A$ that contains $\mathfrak a$ and ideals in the quotient ring $A/\mathfrak a$, but never see the isomorphism above, anyone knows how to prove it?
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1 vote
3 answers
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If $f :R \to S$ is an isomorphism from ring $R$ to ring $S$, both with identity, does $f(1_R) = 1_S$?

My attempted proof: Let $a \in R$, then $f(a) = f(a \cdot 1_R)$ = $f(a) \cdot f(1_R)$, and for $f(a) \ne 0$, it gets canceled, so $1_S = f(1_R)$ Is this correct? Thanks.
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2 votes
1 answer
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Proof of Dedekind-Kummer theorem

I will write the statement and then ask my query about part of the proof. Statement Let $p$ be a rational prime. Let $K=\mathbb{Q}(\theta ) $ be a number field where $\theta $ is an algebraic integer. ...
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  • 1,530
3 votes
2 answers
109 views

isomorphism $\psi$ between quotient rings

Show that there is an isomorphism $\psi$ between the rings $R_1 := \mathbb{Q}[r,s]/(r+s-1, r-r^2, s-s^2)$ and $R_2 := \mathbb{Q}[y]/(y^2-1)$ so that $\psi([r]) = [(1+y)/2]$ and $\psi(s)=[(1-y)/2].$ I ...
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1 vote
0 answers
85 views

Show that $\text{Frac}(\mathbb{Z}(\sqrt{2}))$ is isomorphic to $\mathbb{Q}(\sqrt{2})$

Show that $\text{Frac}(\mathbb{Z}(\sqrt{2}))$ is isomorphic to $\mathbb{Q}(\sqrt{2})$. Here $\text{Frac}(\mathbb{Z}(\sqrt{2}))$ is the fiel of fractions of $\mathbb{Z}(\sqrt{2})$. I already proved ...
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  • 1,842
1 vote
1 answer
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$\mathbb{R}[x,y]/(x^3,y) \cong \mathbb{R}[x,y]/(x^2,y^2)$

I have to prove this ring isomorphism as an exercise. What I can say is that $\mathbb{R}[x,y]/(x^3,y)$ is made of the polynomials of the form $a_0 + a_1 \bar x + a_2 \bar x^2$ (with $\bar x^3 = 0$) ...
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  • 1,463
1 vote
1 answer
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Show isomorphism between localization of rings

I have to show that this two rings are isomorphic $$(\mathbb{Q}[X,Y]/\langle XY-1\rangle)_{\langle X-1,Y-1 \rangle} \simeq \mathbb{Q}[X]_{\langle X-1\rangle}$$ I tried using the fact that $$ \mathbb{Q}...
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  • 405
0 votes
1 answer
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Subring of $\mathbb{Z} \times \mathbb{Z}$

Let $R = \mathbb{Z} \times \mathbb{Z}$ and let $S$ be a subring of $R$ isomorphic to $R$. Show that $S = R$. I know that as $S$ is subring, $S \subseteq R$. But after this I only get that $A$ is ...
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