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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Formula for an inverse isomorphism of commutative rings.

The question I have is styled in the following: "Consider the natural isomorphism of commutative rings:" $\varphi:\mathbb{Z}/(1088) \rightarrow \mathbb{Z}/(17) \oplus \mathbb{Z}/(64)$ Write ...
Cameron's user avatar
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$\mathbb{R}$-Algebra isomorphism $\mathbb{C}[z;\sigma]\cong \mathbb{R} \langle x,y\rangle/(y^2+1,xy+yx)$

Considering the algebra $A=\mathbb{C}[z;\sigma]$ of the skew polynomials, which is like $\mathbb{C}[z]$ but with the multiplication of elements defined the following way: $xb=\sigma(b)x$ and extending ...
Alex A.G.'s user avatar
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Prove that the rings $\mathbb{C}((z;\sigma))$ and $\mathbb{H}((x))$ are not isomorphic.

Prove that the division rings $\mathbb{C}((z;\sigma))$ and $\mathbb{H}((x))$ are not isomorphic. The following statement is needed: \begin{equation} \text{any ring automorphism $\phi: \mathbb{R}((x)) \...
Alex A.G.'s user avatar
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over a Field extension

Let $f(x) = ax^2+bx+c \in \Bbb{R}[x]$ be an irreducible polynomial. Prove that the field $\Bbb{R}[x] \mathbin{/} \langle f(x) \rangle$ is isomorphic to the field of complex numbers. Knowing that $$ \...
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2 answers
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I'm confused by the last step when showing that $\Phi$ is well defined in the proof of 2nd isomorphism theorem for rings?

$S$ is a subring of $R$, $J$ is an ideal of $R$, and $\Phi: (S + J) \to S/(S \cap J)$ where $\Phi(a + b) = a + (S \cap J)$. We need to show that $\Phi$ is well defined. Suppose that $a_1 + b_1 = a_2 + ...
The Big guy's user avatar
3 votes
0 answers
92 views

Isomorphism of quotient containing Cauchy sequences

Let $p$ be a prime number and let $R$ be the ring of Cauchy sequences in $(\mathbb{Q}, |\cdot|_{p})$. Let $\mathfrak{m} \subseteq R$ be the ideal of all sequences converging to $0$, forming a maximal ...
ByteBlitzer's user avatar
2 votes
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60 views

Ringisomorphism between p-adic integers and projective limit.

So I am looking at the following question: we have proven earlier that the map \begin{align*} \left\{ \sum_{\nu = 0}^{\infty} a_{\nu}p^{\nu} \mid 0 \leq a_{\nu} < p \right\} \to \lim_{\substack{\...
ByteBlitzer's user avatar
1 vote
1 answer
84 views

Set of Fixed Elements of a Ring

Let $R$ be a commutative ring with 1, and let $h: R \to R$ be a ring automorphism of $R$. Set $S:= \{x\in R: h(x) = x\}$. That is, $S$ is the set of all elements of $R$ fixed by the automorphism $H$. (...
Important_man74's user avatar
1 vote
1 answer
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Listing methods to prove that two rings are not isomorphic [closed]

Really, this question comes down to listing propeties that are preserved by ring isomorphisms. Off the top of my head, I can think of: cardinality of the ring commutativity the order of elements ...
Adil Raza's user avatar
2 votes
1 answer
64 views

Isomorphism between polynomial rings with several variables

Here is a problem form my abstract algebra course : Let $R = \frac{\mathbb{Q}[u,v,w]}{(u^2v^2 - w^3)}$. Find finitely many monomials $x^{a_j}y^{b_j}$ in $S = \mathbb{Q}[x,y]$ such that $R \simeq \...
EtiBeranger's user avatar
2 votes
1 answer
151 views

Let $p$ be a prime. Show that there exists only two non isomorphic rings with $p$ elements. (Is my solution correct?) [duplicate]

Let $p$ be a prime. Show that there exists only two non isomorphic rings with $p$ elements. My definition of rings does not require that the ring must have an identity $1\neq 0.$ I feel that there are ...
Thomas Finley's user avatar
1 vote
1 answer
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Let $R=\left\{\begin{pmatrix}0&a\\0&b\end{pmatrix}:a,b\in\Bbb Q\right\}$ be a ring and $I=\{A\in R:A^2=0\}$ ideal of $R.$ Show that $R/I\cong\Bbb Q.$

Show that $R=\left\{\begin{pmatrix} 0 & a\\0 & b\end{pmatrix}:a,b\in\Bbb Q\right\}$ is a subring of $M_2(\Bbb Q)$ and $I=\{A\in R:A^2=0\}$ is an ideal of $R.$ Finally, show that $R/I\cong \Bbb ...
Thomas Finley's user avatar
1 vote
2 answers
159 views

Prove that $\Bbb Z [\sqrt 2]\ncong \Bbb Z [\sqrt 3].$ ($\cong$ denotes an isomorphic relation) [duplicate]

Prove that $\Bbb Z [\sqrt 2]\ncong \Bbb Z [\sqrt 3].$ ($\cong$ denotes an isomorphic relation) My solution goes like this: (Here, $I(\phi)$ denotes the kernel of a homomorphism $\phi.$) We have $$\Bbb ...
Thomas Finley's user avatar
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Projections from endomorphism algebra of $G$-equivariant maps

Let $(V,\rho)$ be a representation of finite group $G$. $V$ can be decomposed to the direct sum of isotypic components $$V \cong V_1 \oplus V_2 \oplus \cdots \oplus V_m,$$ where, for each $i$, $V_i$ ...
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If $R= k[x]/(x^3)$, and we have the complex $\cdots \to R \to R \cdots$, where we multiply by $x^2$, then $H_n(C_.) =(x)/(x^2) \cong R/(x)$.

This is from some homological algebra notes. You are given that $R = k[x]/(x^3)$, $k$ is a field, and the complex $$C_.= \cdots R \to R \to R \to \cdots $$ where our differential map $\delta_n$ is ...
dorkichar's user avatar
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Seeking Help with Isomorphism Theorem and Ring Theory

I'm having some trouble solving these tasks: Let R be an integral domain, $a \in R-\{0\}$ an element and $S:=\{a^n\}_{n \geq 0}\subset R$ a) Prove that the mapping $R[X]\rightarrow S^{-1}R$ given by $...
Marco Di Giacomo's user avatar
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Show quotient ring and matrix ring are isomorphic

I need help solving the following problem. Let $F$ be a field and $a\in F$. Show that the quotient ring $F[x]/(x^2-a)$ is isomorphic to the matrix ring $R=\left\{\left(\begin{array}{cc}x&y\\ay&...
user1136316's user avatar
1 vote
1 answer
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Construct a ring isomorphism and its inverse [closed]

I have to construct a ring isomorphism $\mathbb{C}[x,y]/\langle x^2+2xy+y^2-529,x+3y-1\rangle \cong \mathbb{C}\times \mathbb C$ and its inverse. I tried to find a function with $\langle x^2+2xy+y^2-...
tom's user avatar
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Why does Chinese Remainder Theorem imply that respective multiplicative inverse groups are isomorphic?

I came across this result: $\mathbb{Z}/n\mathbb{Z} \cong \mathbb{Z}/n_1\mathbb{Z} \times \mathbb{Z}/n_2\mathbb{Z}$ implies that $\left(\mathbb{Z}/n\mathbb{Z}\right)^{\times} \cong \left(\mathbb{Z}/n_1\...
niobium's user avatar
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Show that any epimorphism from the ring $(Z, +, .)$ onto itself is an isomorphism.

Show that any epimorphism from the ring $(Z, +, .)$ onto itself is an isomorphism. I think by the term epimorphism means an onto homomorphism. Firstly, I assumed, $\phi:Z\to Z$ such that $\phi$ is an ...
Thomas Finley's user avatar
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1 answer
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Calculating the ext group of a cyclic group and an $A$ module.

I'm learning about ext groups right now, but the example I have to work with is confusing. I was wondering if someone could take a look at it, and help fill in the gaps. Let $A=\mathbb{Z}/8\mathbb{Z}$,...
Ty Perkins's user avatar
2 votes
1 answer
111 views

If $A \simeq A_1 \times A_2 \times \cdots \times A_k$, show that $M_n(A) \simeq M_n(A_1) \times M_n(A_2) \times \cdots \times M_n(A_k)$

Let $A,A_1,A_2,\dotsc,A_k$ be rings such that $A \simeq A_1 \times A_2 \times \cdots \times A_k$. Show that $$M_n(A) \simeq M_n(A_1) \times M_n(A_2) \times \cdots \times M_n(A_k)$$ for $n \geq 1$. I ...
Juan Herrera's user avatar
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How do we show $F_2[x]/(x^2 + x) ≅ F_2[x]/(x) × F_2[x]/(x + 1) ≅ F_2 × F_2$?

In the polynomial ring section of the textbook that I am reading, it states: $F_2[x]/(x^2 + x) ≅ F_2[x]/(x) × F_2[x]/(x + 1) ≅ F_2 × F_2$. I know how to show the first pair of isomorphism. Since $(x, ...
L Z's user avatar
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1 answer
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Does the ring homomorphism $f: R_1 → R_2$ mentioned in the First Isomorphism Theorem have to be a bijection?

In the textbook that I am currently learning ring theory from, it states the First Isomorphism Theorem in the following manner: Let $f : R_1 \to R_2$ be a homomorphism of rings. Then the image $\...
L Z's user avatar
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2 votes
0 answers
68 views

Show that $\mathbb Z_{(p)} / J(\mathbb Z_{(p)}) \cong \mathbb Z_p$

For some context, this is one of the exercises on a section about ring homomorphisms. One of the definitions in this exercise is of a local ring (I googled it and it looks like it has a different ...
iwjueph94rgytbhr's user avatar
6 votes
2 answers
838 views

Can two fields which are extensions of one another be non isomorphic

Let $\mathbb L$ and $\mathbb K$ be two fields. We say that $\mathbb L$ is an extension of $\mathbb K$ if there exists a ring homomorphism $\varphi$ from $\mathbb K$ to $\mathbb L$. In that case, $\...
Will's user avatar
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2 votes
0 answers
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Testing isomorphism of simple field extensions

Let $F$ be an arbitrary field, $f$ and $g$ irreducible polynomials over $F$. Consider the fields $F[x]/(f(x))$ and $F[y]/(g(y))$. Is there an algorithm to check whether these extensions are isomorphic?...
Amateur_Algebraist's user avatar
1 vote
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Isomorphism between quotients of two variable formal power series ring

$k$ is a field whose characteristic is not $2$, and $f(x,y)=x^2-y^2, g(x,y)=x^2+x^3-y^2$. Exercise. Show that $k[[x,y]]/(f)\simeq k[[x,y]]/(g)$. So far, I've shown the following. $k[[x,y]]$ is an ...
isz's user avatar
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1 answer
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On quotients of subset rings over infinite sets

$\newcommand{\S}{\mathcal{S}}$ For a set $X$ we definte the subset ring $\S_X$ as the set of subsets of $X$ equipped witht the two operations $$\begin{align} A + B &:= (A\setminus B) \cup (B \...
Gentleman_Narwhal's user avatar
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Prove that $\mathbb{Q}[x]/\langle 3+3x+x^2\rangle \cong \mathbb{Q}(i\sqrt{3})$

If $\mathbb{Q}(i\sqrt{3})=\left\{a+b(i\sqrt{3}) \:\vert\: a,b \in\mathbb{Q}\right\}$ then $\mathbb{Q}[x]/\langle 3+3x+x^2\rangle \cong \mathbb{Q}(i\sqrt{3})$. An earlier part of this problem asks to ...
isaac098's user avatar
1 vote
1 answer
56 views

Decomposition of the field of Laurent formal power series in a polynomial ring and a formal power series ring

Can we decompose the field of formal Laurent series as $$\mathbb C((t))\cong \mathbb C[t^{-1}]\times\mathbb C[[t]]$$ as vector spaces over the field of complex numbers? The map $$\sum_{i\in\mathbb Z}\...
Flavius Aetius's user avatar
1 vote
2 answers
96 views

Isomorphism between quotient polynomial rings

Our professor has given us the following isomorphism (fractions denote quotients): $$\frac{\Bbb{Z}[\sqrt{-3}]}{(2+\sqrt{-3})}\cong\frac{\frac{\Bbb{Z}[X]}{(X^2+3)}}{\frac{(2+X,\,X^2+3)}{(X^2+3)}}.$$ ...
nuuusxd's user avatar
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1 vote
2 answers
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Isomorphism between $R/P$ and $R_P/PR_P$

Let R be a commutative ring, and P a prime bilateral ideal. Let $R_P = \left\{ \frac{a}{s} \ \middle\vert \ a \in R, \ s \not \in P \right\}$ and $PR_P = \left\{ \frac{a}{s} \ \middle\vert \ a \in P, \...
Rararat's user avatar
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1 answer
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show $\mathbb{Q}[t]/(t^2-1)\cong\mathbb{Q}\times\mathbb{Q}$

I want to show $\mathbb{Q}[t]/(t^2-1)\cong\mathbb{Q}\times\mathbb{Q}$ My proof is as follows; First, take $\varphi$ as follows \begin{array}{rccc} \varphi \colon &\mathbb{Q}[t] &\...
fxxxxx's user avatar
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1 vote
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Do isomorphisms of quotients of polynomial rings by monomial ideal preserve the constants?

Let $R \neq \mathbf{0}$ be a commutative ring with identity $1$ and consider the polynomial rings $S = R[x_1, \dots, x_n], T = R[y_1, \dots, y_m]$. Furthermore let $I \subseteq S, J \subseteq T$ be ...
Felix F's user avatar
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1 vote
1 answer
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What is the monoid ring $K[(\Bbb{N}, \text{lcm})]$ isomorphic to?

$(d\mid \cdot)(c\mid \cdot) = (\text{lcm}(d,c) \mid \cdot)$ where $(n\mid x) \in \{0,1\}$ is whether (1) or not (0) $n$ divides $x \in \Bbb{Z}$. Thus, we can linearly extend all formall $K$-linear ...
Daniel Donnelly's user avatar
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55 views

Confusion between Set Equality and Isomorphism Proof Methods when dealing with Fields of Quotients

The common methods for proving $A = B$ and $A \approx B$ are very different. To prove a set equality $A = B$, the common method is to show that $A \subseteq B$ and $B \subseteq A$. For the former, we ...
adam dhalla's user avatar
5 votes
2 answers
504 views

Axiomatic definition of the complex numbers

The real numbers may be defined axiomatically as a complete ordered field. This description characterises them up to isomorphism. Question: is there a similar way to define the field of complex ...
Joe's user avatar
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Let $\psi:\mathbb Z[x] \rightarrow \mathbb R$ be the homomorphism defined by $\psi(p(x))=p(\sqrt3)$.

Let $\psi:\mathbb Z[x] \rightarrow \mathbb R$ be the homomorphism defined by $\psi(p(x))=p(\sqrt3)$. a) Prove that the kernel of $\psi$ is a principal ideal. b) Find the subring $S$ of $\mathbb R$ ...
JAISON ALEXANDER MUNOZ HORMIGA's user avatar
2 votes
2 answers
151 views

Isomorphism Quotient Polynomial Rings 2 Var

Examine Whether: $$ \frac{\mathbb{R}[x,y]}{<x^2-y^2-1>} \cong \frac{\mathbb{R} [x,y]}{<xy -1>} $$ Background: 2nd year math undergrad Currently doing introductory ring theory What I Know\...
Madhav10612's user avatar
2 votes
1 answer
55 views

$R$ be a ring such that every subgroup of $(R,+)$ is again an ideal of $(R,+,*)$. Then $R\cong \mathbb Z_n$.

Aluffi's Algebra Chapter 0 chapter 3- section 3 problem 3.4. Let $R$ be a ring such that every subgroup $I$ of $(R,+)$ is again an ideal of $(R,+,*)$. Then $R\cong \mathbb Z_n$ where $n$ is the ...
Micheal Brain Hurts's user avatar
1 vote
3 answers
140 views

Let $p$ be a prime $\equiv 1 \pmod 4$. Prove that $\Bbb{Z}[i]/(p)$ $\cong$ $\Bbb Z_p \times\Bbb Z_p$ [closed]

I tried to construct a ring homomorphism from $\Bbb{Z}[i]$ to $\Bbb Z_p \times\Bbb Z_p$ but it turns out $f$($a+bi$) = $(a,b)$ is not a homomorphism because it doesn't split over multiplication. I ...
Nel's user avatar
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2 votes
1 answer
85 views

Adjoining $i$ to $\mathbb{C}$

Let $I$ be an ideal of a ring. I know that $I$ is maximal if and only if $R/I$ is a field. There's an example I'm working with, $(x^2+1) \subset \mathbb{C}[x]$ which is clearly not maximal because $(x^...
Ty Perkins's user avatar
4 votes
2 answers
87 views

Isomorphisms in Product Rings

I'm working on a problem in Michael Artin's Algebra that asks: Describe the ring obtained from the product ring $\mathbb{R} \times \mathbb{R}$ by inverting the element $(2,0)$. In the last problem, I ...
Ty Perkins's user avatar
1 vote
1 answer
120 views

What does the correspondence of ideals containing $I$ and ideals of $R/I$ tell us about computing quotients?

$\newcommand{\Z}{\mathbb{Z}}$ Here is the situating example. Example: Let $R = \Z[x]$ and consider the ideal $I = (3,x^2-13) = \{3f+(x^2-13)g \mid f,g \in R\}$. The goal is to compute the quotient $R/...
Irving Rabin's user avatar
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0 votes
1 answer
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Relaxing the hypotheses of the theorem that any complete ordered field is isomorphic to the real numbers

One of the central theorems in real analysis states that any complete ordered field $X$ is isomorphic to $\mathbb R$. Here, I mean "complete" in the sense of the least upper bound property, ...
Joe's user avatar
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Question about algebraically independent isomorph to $A[X_1,\ldots,X_n]$

(we denote by $X_1,\ldots,X_n$ indeterminates) Let $A$ be subring of $R$ (both commutative). The elements $x_1,\ldots,x_n \in R$ are called algebraically independent over $A$, if for all Polynomials $\...
willix's user avatar
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Is $\mathbb{Q}[1]$ isomorphic to $\mathbb{Q}$?

If I am not mistaken, $\mathbb{Q}[1] \cong \mathbb{Q}/(f(X))$ denotes the ring of polynomials with rational coefficients evaluated in $1$ ($1$ being the root of the polynomial $f(X)$ as well). Could ...
lvo0224's user avatar
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1 vote
0 answers
77 views

Prove that the rings $\frac{\Bbb F_3[X]}{(X^3+X^2+2)}$ and $\frac{\Bbb F_3[X]}{(X^3+2X+2)}$ are isomorphic

Prove that the rings $\frac{\Bbb F_3[X]}{(X^3+X^2+2)}$ and $\frac{\Bbb F_3[X]}{(X^3+2X+2)}$ are isomorphic. I tried to construct an explicit isomorphism using a similar technique as used in this and ...
Sayan Dutta's user avatar
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1 vote
1 answer
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Proof of Kummer's Theorem in Janusz's Algebraic Number Fields

There is a theorem in Janusz's Algebraic Number Fields stated as follows: Kummer's Theorem: Let $R$ be a Dedekind ring with quotient field $K$ and $R'$ the integral closure of $R$ in a finite ...
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