Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [ring-isomorphism]

The tag has no usage guidance.

2
votes
2answers
64 views

Finding all polynomials $p(x)$ such that $ \mathbb{Q}[x]/p(x) \simeq \mathbb{Q}(A)$ for fixed $A$ algebraic

As an example, if we put $F =\mathbb{Q}(\sqrt{2}, \sqrt{3}, ... , \sqrt{n})$, $F$ is the splitting field of $p(x)$ so that we can write $$ p(x) = (x^2 - 2)(x^2 - 3)\cdots (x^2 - n). $$ Question: if ...
1
vote
1answer
41 views

Is the following ring isomorphism true?

In recent few days, I have been trying to develop an intuitive idea for quotient rings and what I've learnt from my explorations is that if you want to quotient a ring by a principal ideal, you can ...
-1
votes
2answers
47 views

Abstract Algebra: Quotient ring, ideal, and isomorphism

I need help with the following exam exercise, my teacher didn’t post the answer and I can’t manage to solve it. In $ A=\mathbb{Z}[i]=\{a+bi \ : \ a,b \in \mathbb{Z}\} $ we consider $a=7+56i; \ b=3+...
0
votes
3answers
41 views

Finding an isomorphic subring of matrices

I'm struggling a fair amount with this exercise: Find a subring of $M(2,\mathbb{Q})$ which is isomorphic to a) $\mathbb{Q}$ x $ \mathbb{Q}$ b) $\mathbb{Q}$ c) $\mathbb{Q}[x]$/$x^2$ Now I know a ...
3
votes
1answer
66 views

Showing that $\frac{\mathbb{R}[x]}{\langle x \rangle}$ and $\frac{\mathbb{R}[x]}{\langle x-1 \rangle}$ are not isomorphic as $\mathbb{R}[x]$ modules.

I'm trying to solve an exercise which asks me to prove that $\frac{\mathbb{R}[x]}{\langle x \rangle}$ and $\frac{\mathbb{R}[x]}{\langle x-1 \rangle}$ are isomorphic as rings, but not as $\mathbb{R}[x]$...
1
vote
2answers
32 views

Product ring isomorphism from example 11.6.3 in Artin's Algebra

I am currently reading chapter 11.6 in Artin's Algebra on Product Rings. There's a proposition that says if $e$ is an idempotent element of a ring $S$ and $e' = 1 -e$ then $S \cong eS \times e'S$. I ...
3
votes
2answers
46 views

Prove or disprove ring $\mathbb{C}\times \mathbb{C}$ and ring Quaternion $H$ are isomorphic

My attempt: let $f$ ($w$+$x$i+$y$j+$z$k) $=$ ($w$+$x$i,$y$+$z$i) then I tried proving it is a homomorphism. $f$ is a homomorphism under addition but fails to be a homomorphism under multiplication. ...
2
votes
2answers
47 views

Isomorphisms of $\mathbb{R}[A]$ where $A$ is a $2\times2$ real matrix.

I'm trying to answer a question in which I'm supposed to show that if $A$ is a $2\times2$ real matrix then $\mathbb{R}[A]$ (the polynomials in $A$ with real coefficients) is isomorphic to one of: $$\...
2
votes
2answers
43 views

Algebraic structure, division ring

In the set $\mathbb{R}\times \mathbb{R}^3$ addition is defined by components. We define multiplication $*$ by $$(\lambda,\mathbf{x})*(\mu,\mathbf{y})=(\lambda\mu - \mathbf{x}\cdot \mathbf{y}, \lambda \...
0
votes
3answers
61 views

Need help to construct a ring isomorphism from $\mathbb{Z}_3[\sqrt2]$ $\rightarrow $ $\mathbb{Z}_3[i]$

I've looked at this problem for some time now and am confused as to where to start. I know that to prove these rings are isomorphic I must to construct a bijective function $\phi$ such that: $\phi(a +...
1
vote
2answers
53 views

Proving 2 quotient rings are isomorphic

So I want to show $\mathbb{R}[x]/((x-r)^2)$ is isomorphic to $\mathbb{R}[x]/(x^2)$ where $r \in \mathbb{R} $. I thought of constructing a ring homomorphism $\phi : \mathbb{R}[x] \to \mathbb{R}[x]/(x^2)...
2
votes
0answers
45 views

What's the point of studying isomorphisms?

I'm trying to learn about groups/rings and the concept of isomorphisms appears everywhere. I understand that an isomorphism between groups/rings shows that arithmetic in both structures is essentially ...
1
vote
1answer
38 views

Isomorphisms for rings modulo ideals

Let the following ring be given: $\mathbb{Z}[\sqrt{-3}] := \{a+b\sqrt{-3}: a,b\in\mathbb{Z}\}$. I was wondering what the following quotients would look like, given the ideals. For the ideal $(\sqrt{-...
0
votes
0answers
21 views

Showing that $(I+J)/(I\cap J)\cong (I+J)/I\times (I+J)/J$.

Show that $(I+J)/(I\cap J)\cong (I+J)/I\times (I+J)/J$, where $I,J$ are ideals in a commutative ring $R$ with identity. What I did was first show that $\phi: I+J\to (I+J)/I\times (I+J)/J$ defined by $...
1
vote
1answer
69 views

What does $φ(a) = a$ mean in this statement?

Let $F$ be a field and let $φ:F[x] \to F[x]$ be an isomorphism such that $φ(a)=a $ for every $a$ in $F$. Prove that $f(x)$ is irreducible in $F[x]$ if and only if $φ(f(x))$ is. [Hint: First prove that ...
4
votes
2answers
126 views

Are the rings $\mathbb{R}^2$ and $\mathbb{R}^3$ isomorphic?

Are the rings $\mathbb{R}^2$ and $\mathbb{R}^3$ isomorphic, where $\mathbb{R}^2=\mathbb{R}+\mathbb{R}$ and is the set of all pairs $(a,b)$ with $a,b \in \mathbb{R}$, and $\mathbb{R}^3=\mathbb{R}+\...
2
votes
1answer
53 views

Are multiplicative monoids of different rings isomorphic?

I have some algebraic problem and hope some of you can help me! The problem is about rings and their multiplicative monoids (semi-groups with neutral element $e$). So $M(\mathbb{Z}), M(\mathbb{Z}_2[x])...
2
votes
1answer
38 views

Group ring isomorphic to matrix ring

I want to figure out whether $R$ is isomorphic to $S$, where $R = \mathbb{R}[G]$, where $G = \mathbb{Z}/2 \times \mathbb{Z}/2$, and $S = M_4(\mathbb{R})$. It seems that they might not be isomorphic, ...
0
votes
0answers
26 views

Showing that three rings are isomorphic

Let $C_3 = \langle a | a^3 = e \rangle$ and let $R=(\mathbb{Z}/2)[C_3]$ be the group ring of $C_3$ with $\mathbb{Z}/2$ coefficients. Let $S = (\mathbb{Z}/2)[y]/(y^3-[1])$ and let $T = \mathbb{Z}[x]/(...
2
votes
1answer
60 views

Proof: $ { \mathbb{Q}[x] } / (x^2 +3x +1) \cong \mathbb{Q}(\sqrt{5}) $

I have a problem with proving this: $$ \mathbb{Q}[x] / (x^2 +3x +1) \cong \mathbb{Q}(\sqrt{5}) $$ My observations: In $$\mathbb{Q}[x] / (x^2 +3x +1)$$ $$x^2 \equiv -1 -3x$$ so $$\forall f \in \...
0
votes
1answer
58 views

Isomorphic ring $\mathbb{Q}[x]/(x^3+x) \cong \mathbb{Q} \times \mathbb{Q}[x]/(x^2+1)$

I want to prove an isomorphism of the form $\mathbb{Q}[x]/(x^3+x) \cong \mathbb{Q} \times \mathbb{Q}[x]/(x^2+1)$. I want to use the Chinese Remainder Theorem $\mathbb{Q}[x]/(x^3+x) = \mathbb{Q}[x]/(x(...
2
votes
1answer
75 views

Fundamental ismorphism theorem

I don't understand how to apply the fundamental isomorphism theorem to polynomial quotient rings. For example is the ring $\mathbb C[X,Y,Z]/\langle X^2-Z,XZ-Y^3\rangle$ isomorphic to $\mathbb C[X,Y]/\...
0
votes
1answer
92 views

Showing isomorphism of Quotient Ring to direct product of Complex numbers

So I need to prove $$\mathbb{C}[x]/(x^3+1)$$ is isomorphic to $\mathbb{C} \times \mathbb{C} \times \mathbb{C}$, where $\mathbb{C}$ is the field of complex numbers. Based on an example in the book I ...
0
votes
2answers
92 views

Defining isomorphism of rings

I'm having some difficulty with the following problem: Define an isomorphism of rings $$f: \mathbb{Q} [T]/(T^2-d) \rightarrow \mathbb{Q}(\sqrt{d})$$ and an isomorphism $$g: \mathbb{R} [T]/(T^2+1) \...
3
votes
0answers
56 views

Ring isomorphic to $\mathbb{Z}[1/3]/(5)$?

I am supposed to find a ring that is isomorphic to $\mathbb{Z}[1/3]/(5)$ using an isomorphism theorem. My guess is that $\mathbb{Z}[x]/(5,1+3x)$ is isomorphic to this and in turn I think that $\mathbb{...
0
votes
2answers
58 views

Showing that if $\Bbb Z/ab\Bbb Z$ is isomorphic to $\Bbb Z/a\Bbb Z \times \Bbb Z/b\Bbb Z$, then $gcd(a,b)=1$ necessarily.

I've been attempting this every which way and I seem to be going in circles at this point. I feel like the solution is probably very simple, but for whatever reason I am not seeing it. This isn't a ...
0
votes
3answers
16 views

Relationship between a polynomial and its constant term (ring)

There is an example illustrating ring isomorphism that surprises me so much. It essentially talks about the relationship between a polynomial and the constant term: Consider the ring $\mathbb{Q}[t]$...
1
vote
2answers
30 views

Showing this map is injective

Define a map $$\phi : \mathbb{Q}[x] \rightarrow \mathbb{Q}[x]$$ by $\phi(g(x)) = g(x-2)$ It is straightforward to show that this is a homomorphism, but I'm having trouble showing it's surjective &...
0
votes
0answers
21 views

Showing the factor ring $\mathbb Z[x]/\langle x^2 + 1\rangle$ is isomorphic to the Gaussian integers. [duplicate]

I'm hoping for feedback on my understanding of how to approach this question. Firstly to recall; $$R = \mathbb Z[x]/\langle x^2 + 1\rangle = \{ l(x) + \langle x^2 + 1\rangle | l(x) = 0 \text{ or } ...
2
votes
1answer
52 views

Seeking an isomorphism from $\langle\mathbb{Q},=,<\rangle\to\langle\mathbb{Q}\cap((0,1)\cup(2,3)),=,<\rangle$

As the question suggests, I am seeking an isomorphism from $$\langle\mathbb{Q},=,<\rangle\to\langle\mathbb{Q}\cap((0,1)\cup(2,3)),=,<\rangle.$$ I know for example that $$\frac{x}{2+2|x|}+\frac{1}...
2
votes
1answer
31 views

How to prove isomorphism for rings in Galois Theory?

The question is for a module on Galois theory and I haven't a clue where to start: Take $f \in \mathbb{K}[x]$ with $deg(f)>0$. Show that the map $\lambda \mapsto \lambda+<f>$ is an injective ...
2
votes
0answers
49 views

Extending Isomorphisms

Let $E$ and $F$ be fields, and define the map $*^{\sigma} : F[x] \rightarrow F'[x]$, which simply applies $\sigma$ to the coefficients of a polynomial $f(x)$. It is an isomorphism of rings. Let $\...
1
vote
1answer
39 views

Show that $\mathbb{Q}[X]/\langle X^3+X\rangle\cong \mathbb{Q} \times \mathbb{Q}[i]$

I've seen that $\mathbb{R}[X]/\langle X^2+1\rangle\cong \mathbb{C}$ with the evaluation in i. I tried to replicate the idea, but I don't end up having that $\ker(f)=\langle X^3+X\rangle$, but $\langle ...
0
votes
2answers
47 views

Show that there is no isomorphism

I want to show that $\mathbb R[x]/(x-a)^2$ is not isomorphic to $\mathbb R\times \mathbb R$. If we take $a = b = x-a$ in $\mathbb R[x]/(x-a)^2$, then $ab=0$ and $f(ab)=0$. If there is isomorphism than ...
2
votes
2answers
37 views

Homomorphism in Rings

I have the following true/false statement from a worked example: If $F_1, F_2$ are fields and $\phi: F_1 \rightarrow F_2$ is a homomorphism, then $\phi$ is either the zero map or an isomorphism. ...
1
vote
1answer
56 views

If $A= \mathbb{Z}[X]/(X^n+1)$, is it true that $A/mA \cong \mathbb{Z}_m[X]/(X^n+1)$?

Let $R= \mathbb{Z}[X]/(X^n+1)$ for some sufficiently large $n$. For $q \geq 2$, I want to show that $R/qR \cong \mathbb{Z}_q[X]/(X^n+1)$. I've tried to prove it, but I dont know the construction of $...
4
votes
0answers
75 views

Lifting $\overline{\mathbb{Q}}_{p}$-algebra isomorphism to $\overline{\mathbb{Z}}_{p}$-algebra isomorphism

Let $A$ and $B$ be two $p$-torsion free $\overline{\mathbb{Z}}_p$-algebras of finite type and suppose that $A\otimes_{\overline{\mathbb{Z}}_{p}}\overline{\mathbb{Q}}_{p}$ and $B\otimes_{\overline{\...
4
votes
2answers
209 views

Show that two rings of matrices are not isomorphic

Let $p$ be a prime number and $$ A_p = \{\left( \begin{matrix} a & bp \\ b & a \\ \end{matrix} \right)|\ a, b \in \mathbb{Z} \} $$ Show that $A_2$ and $A_3$ are not isomorphic. ...
4
votes
2answers
80 views

$A\times B\simeq C\times D \implies A\simeq C\text{ or } A\simeq D$

Let $A,B,C,D$ be integral domains such that $A\times B$ is isomorphic to $C\times D$. Prove that $A$ is isomorphic to either $C$ or $D$. Normally I include my thoughts about the problem, but I don't ...
0
votes
0answers
59 views

How many rings are there of order 4 up to isomorphism [duplicate]

How many rings of order 4 can exist upto isomorphism there are only two groups $Z_4$ and $K_4$ upto isomorphism where $Z_4$ is abelian and $K_4$ is non-abelian.To be a ring that must be a abelian ...
1
vote
0answers
32 views

How to Prove Isomorphism Between Two Quotient Fields?

I have following two field and would like to prove those two are isomorphism as a field. $F_1 = \Bbb Z_3[x] /( x^3+x^2+2)$ $F_2 = \Bbb Z_3[x]/(x^3 + 2x+2)$ Then I might have to prove 1) ...
1
vote
2answers
37 views

Check whether two given given rings are isomorphic

Examine with justification whether $\Bbb Q[X]$/$[X^2-X]$ and $\Bbb Q$$\times$$\Bbb Q$ are ring isomorphic. My approach: say if we take any polynomial $aX^3+bX^2+cX+d$ in $\Bbb Q[X]$, so in $\Bbb Q[X]...
1
vote
1answer
24 views

Prove that if $x^2 +\bar ax +\bar 1$ has a double root, then $(\mathbb Z/5)[x]/I$ $\cong$ $(\mathbb Z/5)[x]/(x^2)$ where I = $x^2 +\bar ax +\bar 1$

I'm pretty sure that I am meant to use the evaluation map, but I'm confused about how to write a homomorphism that doesn't have a kernel of $(x-c)$ where $(x-c)^2$ is $x^2 +\bar ax +\bar 1$
1
vote
1answer
100 views

Why is $\mathbb{Z}[x]/(2, (x^3+1))\cong \mathbb{F}_2[x] /(x^3+1) $?

I am trying to understand Yuchen's answer to my other question here. The first line is $$\mathbb{Z}[x]/(2, (x^3+1))\cong \mathbb{F}_2[x] /(x^3+1)$$ I would like to understand please two things: $...
1
vote
2answers
58 views

Why is there no subring of the integers modulo 29 isomorphic to the integers modulo 8?

I understand that for rings R and S, S contains a subring isomorphic to R if and only if there is an injective ring homomorphism from R to S. But I am unsure of how to proceed from this definition? ...
0
votes
2answers
62 views

How to show that $\mathbb{Z}/13\mathbb{Z}$ is isomorphic to $\mathbb{Z}[i]/(3-2i)$?

I know that there is a unique morphism between $\mathbb{Z}$ and $\mathbb{Z}[i]/(3-2i)$ defined by $f(x)=x+(3-2i)$. I want to use the first isomorphism theorem, but i do not know how to show that $f$ ...
0
votes
1answer
52 views

Finding a homomorphism from a polynomial to matrix

I have to show that $$R_k\cong\mathbb{Z}[X]/(X^2-k)$$ for $R_k= \left\{\left( {\begin{array}{cc} a & b \\ kb & a \\ \end{array} } \right)\ \middle| \ a,b\in \mathbb{Z} \right\} , k\...
2
votes
2answers
50 views

$ R \cong R s $ as rings if and only if $ s $ is invertible

Let $ R $ be an integral domain and let $ s \in R $ be nonzero. Then why is it true that $ R \cong R s $ if and only if $ s $ is invertible? I can see that it is true for all nonzero $s$ if we ...
2
votes
1answer
30 views

Show that the ring isomorphism $\phi :R[x]\to S[x]$ is well-defined.

Could someone please verify whether my answer is too simple or needs more detail or something is wrong? Let $\phi_{1}:R\to S$ be a ring isomorphism. Show that the ring isomorphism $\phi_{2} :R[x]\...
1
vote
1answer
252 views

For ring homomorphism $\phi:R\to S$, prove that $\phi$ is an isomorphism iff $\phi$ is onto and $ker(\phi)=\left \{0_{R} \right \}$.

Could someone please verify whether my solution is okay? I understand this proposition is considered trivial, but I am not sure whether my proof is correct. For ring homomorphism $\phi:R\to S$, ...