Questions tagged [ring-homomorphism]

For questions about ring homomorphisms, a function between two rings which respects the structure.

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Show that $\text{ker}(\alpha) \subseteq \text{ker}(\beta\alpha)$. If $\beta$ is one-to-one, show that $\text{ker}(\beta\alpha) =\text{ker}(\alpha) $

Let $\alpha : R \rightarrow S$ and $\beta : S \rightarrow T$ be ring homomorphisms. Show that $\text{ker}(\alpha) \subseteq \text{ker}(\beta\alpha)$. If $\beta$ is one-to-one, show that $\text{ker}(...
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1answer
109 views

Find all ring homomorphisms from $\mathbb{Z} \rightarrow \mathbb{Z}_m$

I want to find all rings homomorphisms from: i) $$\mathbb{Z} \rightarrow \mathbb{Z}_m$$ ii) $$\mathbb{Z}_m\rightarrow \mathbb{Z}$$ iii) $$\mathbb{Z}_n \rightarrow \mathbb{Z}_m$$ I don't know how to ...
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1answer
49 views

Show that $x^2+\mathbf{1}=\mathbf{0}$ has two solutions in this ring homoporphism

I have the following question here: Given $a,b\in\mathbb{R}$, let $$[a,b]=\begin{pmatrix}a & -b \\ b & a \end{pmatrix}\in M_2(\mathbb{R})$$ Define $F=\{[a,b]\in M_2{(\mathbb{R})}|a,b\in \...
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Is there a non-constant function $f$ such that $f(n) - 1 = 0$ for all $n \in \mathbb{N}$?

So I've been given rings $R = Map(\mathbb{R},\mathbb{R})$ and $S = (a_{n})_{n \geq 0}$ such that $a_{n} \in \mathbb{R}$ and the ring homomorphism $$ \phi: R \rightarrow S\\ f \rightarrow (f(n))_{n} $$ ...
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First isomorphism theorem with sequences

I have the following here: Let $R=\text{Map}(\mathbb{R},\mathbb{R})$ with addition and multiplication defined, as usual, by $(f+g)(x)=f(x)+g(x)$ and $(f \cdot g)(x)=f(x)g(x)$. Let $S$ be the ring of ...
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1answer
56 views

How to show this ring homomorphism is surjective?

So I've been given the following R = Map(ℝ, ℝ) with addition and multiplication defined by $(f +g)(x) = f(x)+g(x)$ and $(f ·g)(x) = f(x)g(x)$. Let S be the ring of sequences $(a_{n})_{n}≥0$ with ...
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Ring homomorphism $\varphi : m\mathbb Z \to \mathbb Z _{\frac{n}{m}}$

This is what I have to prove: The map $\varphi : m\mathbb Z \to \mathbb Z _{\frac{n}{m}}$ where $m \mid n$ given by $\varphi(x) =\overline{\left(\frac{x}{m}\right)}$ is a (not necessarily unital) ring ...
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1answer
61 views

Show that $R/I$ is isomorphic with $\mathbb{Z}/5\mathbb{Z} \oplus \mathbb{Z}/5\mathbb{Z}$

$R=\{a+ib:\ a,b\in\mathbb{Z}\}$ $I=\{a+ib:\ 5\mid a\text{ and }5\mid b\}$ Ok so I wanted to create homomorphism from $R$ onto $\mathbb{Z}/5\mathbb{Z} \oplus \mathbb{Z}/5\mathbb{Z}$. $\phi(a+ib)=([a],[...
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Are there any useful generalizations of homomorphisms that can apply to a graded valuation ring?

I have two graded valuation rings, $R$ and $S$, that I want to relate by a function $f:R\rightarrow S$. The problem is that $f$ is only a group homomorphism under addition for two elements of $R$ ...
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1answer
46 views

Localization of a ring isomorphism and algebras

Let $\phi:R\rightarrow S$ be a homomorphism and $S$ an $R-$algebra. Moreover, let $T\subseteq R$ be a multiplicative group. Prove that $T^{-1}S$ is a $T^{-1}R$ algebra and that $T^{-1}S\simeq \phi(T)^{...
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1answer
82 views

Is the map $R^\text{op} \to \text{End}_R(M)$ defined by $a \mapsto (x \mapsto ax)$ valid?

Let $R$ be a not-necessarily commutative ring, and suppose $M$ is left $R$-module. Let $R^\text{op}$ consist of elements written as $a^\text{op}$, where as an element $a^\text{op} = a$ for all $a\in R$...
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1answer
26 views

Does an $R$ algebra always contain $R$?

In an associative algebra with unit over a commutative ring $R$ it's true that $R$ is inside the algebra? And, is $1$ is the unit in the algebra, is this inclusion $R\cdot1$?
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1answer
21 views

Show there exists an injective ring homomorphism that makes the diagram commute.

I am taking some practice exams to relearn algebra. I'm having trouble with questions regarding showing there exists or doesn't exist a ring homomorphism. Here's a particularly scary-looking one I can'...
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Ring homomorphisms between convolution rings $R_{2^n}$ and $R_{2^k}$ for $k=1,2,\ldots,n-1$.

I'm currently studying weaknesses in the algebraic structure of the convolution rings used in the NTRU cryptosystem. Moving forward, let $R_q:=(\mathbb{Z}/q\mathbb{Z})[X]/\langle X^N-1\rangle$ with $N$...
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Quotients of multivariate polynomial rings - is $k[x][y]/(y-x^2) \cong k[y][x]/(y-x^2)$?

This question is motivated by exercise 1.1 in Hartshorne Algebraic Geometry. One has to prove that $k[x,y]/(y-x^2)$ is isomorphic to a polynomial ring in one variable. I know that this can be done by ...
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35 views

Determine the ring homomorphism with domain $\mathbb{Q}?$ [duplicate]

Determine all the ring homomorphism from $\mathbb{Z}$ to $\mathbb{Q}$ and $\mathbb{Q}$ to $\mathbb{Z}$ My attempt : Ring homomorphism from $\mathbb{Z}$ to $\mathbb{Q}$ Here $\mathbb{Z}$ is cyclic ...
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1answer
95 views

Are $\mathbb{Q}(π)$ and $\mathbb{Q}(π^2)$ the same field?

I've tried to check if $\mathbb{Q}(π)$ and $\mathbb{Q}(π^2)$ are equal fields. I know there exists a field isomorphism between $\mathbb{Q}(π)$ and $\mathbb{Q}(π^2)$ since $π$ and $π^2$ are ...
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30 views

No onto map between the algebras.

I want to prove that there does not exist a surjective group algebra homomorphism from $FS_5$ (the group algebra (or group ring) of the symmetric group, $S_5$, over the field $F$) to $M_2(F)$, where $...
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1answer
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Why does field theory still work with generalized field extensions (embeddings)?

When $E/F$ is a field extension, I've always assumed that $E \supseteq F$. However, there is a more general definition that $E/F$ is a field extension iff $E, F$ are fields and there is an embedding $...
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Does every evaluation map $g: \Bbb{Z}[X,Y]/(\ker \pi) \to \Bbb{Z}$ factor some evaluation map $f:\Bbb{Z}[X,Y] \to \Bbb{Z}$?

Let $\Bbb{Z}[X,Y]$ be the polynomial ring. I know that every evaluation ring hom $f: \Bbb{Z}[X,Y] \to \Bbb{Z}$ is determined by where you send $X$ and $Y$. Let $I = (X^2 - Y^3)$ for example, but it ...
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39 views

When does a monomorphism must be an automorphism?

Say R is a ring, and denote $\varphi :R\rightarrow R$ a one-to-one homomorphism. Consider the case where $\varphi$ is not onto. One example is the mapping $(a_{0}, a_{1}, ...) \mapsto (0, a_{0}, a_{1},...
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63 views

Ring Homomorphism Question

Let $f:F_1 → F_2$ be a ring homomorphism between fields $F_1, F_2$. (a) Show that if $f(1)=0$ then $f=0$. (b) Show that if $f(1)\ne0$ then $f$ is injective. Hi, I'm not too sure how to do this ...
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28 views

Inclusion of ideal in a finite union of ideals [duplicate]

Say $R$ is a ring, and $I_1,...,I_n,J\subseteq R$ ideals, s.t $J\subseteq\bigcup_{i=1}^nI_i$. If there exists $\phi:K\rightarrow R$ homomorphism where $K$ is an infinite field, then there exists an $i$...
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51 views

Proof of Commutativity of the ring of endomorphisms over integers

First I searched for some similar problems and studied some of the given references, but I was not able to understand them. So I pose the question, hopefully to reveal some aspects to a less advanced ...
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1answer
111 views

Ring endomorphism of $p$-adic integers

I am doing an individual study of an abstract algebra for number theory course online. I just started, so I hope my question just note come off as too trivial. The lecture notes state that the ring of ...
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1answer
30 views

Ring homomorphism from $\mathbb{F}[x,y]$ to $\mathbb{F}[u]$

Is it possible to map from $\mathbb{F}[x,y]$ to $\mathbb{F}[u]$ where each monomial $x^ay^b = u^{aw+b}$ for appropriate choice of $w$? In fact this mapping $\phi(x^ay^b) = u^{aw+b}$ seems to be ...
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1answer
31 views

The kernel of an evaluation homomorphism

We let $R$ be a commutative ring with unity, and $R[X_1,\dots, X_n]$, $a=(a_1,\dots, a_n)\in R^n$ and $\phi_a:R[X_1,\dots, X_n]\rightarrow R$, $\phi_a(f)=f(a)$. I want to show that $\ker(\phi_a)=(X_1-...
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18 views

Evaluation homomorphism with several indeterminates

Let $F$ be a subfield of a field $E$, let $\alpha$ be any element of $E$, and let $x$ be an indeterminate. The map $\phi_{\alpha} : F[x] → E$ defined by $\phi_{\alpha}(a_0+a_1x+...+a_nx^n)=a_0+a_1\...
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27 views

Isomorphism of quotient rings with Cartesian product

If $I$ and $J$ are ideals in $R$ and $S$ respectively, how do I show that $(R\times S)/(I\times J)\cong(R/I)\times(S/J)$? I started by showing that $I\times J$ is an ideal in $R\times S$ but am unsure ...
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1answer
122 views

Considering there exists a homomorphism between two rings show that a condition must exist.

$R$ is defined as \begin{bmatrix}a&b\\b&a\end{bmatrix} such that $a,b \in \mathbb{Z}$ where R is a subring of all 2-by-2 matrices with integer inputs. There exists a homomorphism $θ :R→Z$ such ...
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1answer
48 views

Prove $\text{Hom}(K,K)=\{0,id_K\}$ for $K=\mathbb{Q}$

I've been solving some problems from my Galois Theory course, and I need help with the final step of this one: Prove $\text{Hom}(K,K)=\{0,id_k\}$ for $K=\mathbb{Q}$. The work I did so far: Given $K$ ...
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65 views

Prove that the kernel is a principal ideal [duplicate]

Is the kernel of $\mathbb Z [x]\to \mathbb R, x\mapsto 1/2+\sqrt{2}$ a principal ideal? I proved that $(4x^2-4x-7)$ is a subset of the kernel. Now I need to prove that everything in the kernel lies ...
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1answer
52 views

A problem about ring isomorphism?

problem:let $R$ and $R'$ be commutative rings,and let $I⊆R$ and $I'⊆R'$ be ideals. if $f:R→R'$ is homomorphism with $f(I)⊆I'$,prove that : (i)$f_*:r+I→f(r)+I'$ is well-defined homomorphism $f_*:R/I→R'...
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52 views

The homomorphic images of ring centers are central (Dummit and Foote Exercise 7.3.16 and related question)

Let $\phi: R \rightarrow S$ be a surjective homomorphism of rings. Prove that the image of the center of $R$ is contained in the center of $S$. Which is from Dummit and Foote Exercise 7.3.16. I have ...
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1answer
35 views

On an Isomorphism of Semigroup Rings via Congruence Classes

Let $\mathbb Z_{\geq 0}$ denote the set of non-negative integers. Let $\mathbb Z_{\geq 0}^n$ denote the set of $n$-tuples of non-negative integers. (Theorem 2.1.5, Herzog, 1969) Given a finitely ...
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120 views

A surjective homomorphism which is not an isomorphism for rings (Using basic algebra)

Give an example of a ring $R$ and a surjective homomorphism $R \to R$ that is not an isomorphism. Here is my approach. I know a lemma that says that if $R$ is a Noetherian ring, and $f: R \to R$ is a ...
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65 views

Verify $f: \mathbb{Z} ⟶ \mathbb{Z}_{r} \times \mathbb{Z}_{s}$ given by $a ⟶ (ā_r, ā_s)$ is an epimorphism

Let $r,s \in \mathbb{Z}^+$ such that $(r,s)=1.$ Consider the ring $\mathbb{Z}_{r} \times \mathbb{Z}_{s}$ (with respect to ordinary multiplication and addition, component by component), prove that the ...
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3answers
136 views

Example to show that $\mathbb Z^3$ is not a quotient of $\mathbb Z[X]$

I am asked to prove that $\mathbb{Z}^n$ is not a quotient of $\mathbb{Z}[X]$ for any integer $n \geq 3$ by showing that there is no surjective ring homomorphism $\mathbb{Z}[X] \to \mathbb{Z}^n.$ We ...
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27 views

Show that the following maps are (not) ring homomorphisms. Complex Numbers.

Problem: Show that the following maps are (not) ring homomorphisms (with or without 1): $\phi: \mathbb{C} \to \mathbb{R}^{2\times2}, a +bi \mapsto \left( \begin{array}{rrr}a & b\\ -b & a \\ \...
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1answer
22 views

Prove or confute the follow proposition

I have to prove or confute the follow proposition: Let $F$ a field and $ f: $$\mathbb Z \rightarrow F$ an homomorphism of rings such that $f(1_\mathbb Z ) =1_F$. If $f$ isn't injective $\Rightarrow$ $...
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1answer
37 views

Proving an elementary proposition about modules.

I want to prove the following: Let $R$ be a commutative ring and $M$ is an abelian group then $M$ is an $R$-module iff there exists a ring homomorphism $f: R \rightarrow End_Z(M).$ My professor said ...
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1answer
30 views

If $\phi(a)$ is invertible then $a$ is invertible

Let $R$ and $S$ be rings and $\phi: R \to S$ is a surjective ring homomorphism. If $\phi(a)$ is invertible then $a$ is invertible Is this correct? We know $\phi(1_R)=1_S$ but I am confused thats what ...
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43 views

Let $M$ be a left $R-module$. Prove that the following are equivalent:

Let $M$ be a left $R-module$. Prove that the following are equivalent: (a) M is simple; (b) Every non zero homomorphism $M \to N$ is a monomorphism; I know that a module $M$ is simple if and only if ...
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1answer
40 views

If $f$ is not injective then $F$ is finite.

I have to prove or confute the follow proposition: Let $F$ a Field and $f:\mathbb{Z} \to F$ an homomorphism of rings such that $f(1_\mathbb{Z})=1_F$. Show that if $f$ isn't injective then $F$ is ...
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2answers
78 views

Prove that ${\mathbb{Q}}/{\mathbb{Z}}$ is isomorphic to $\bigoplus_p \mathbb{Z}[p^{\infty}]$

Prove that ${\mathbb{Q}}/{\mathbb{Z}}$ is isomorphic to $\bigoplus_p \mathbb{Z}[p^{\infty}]$ I know we will use $A_p$'s for solution. But i dont know how to do the isomorphism between ${\mathbb{Q}}/{\...
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1answer
19 views

example of a map in a prime ring

Let $R$ be a prime ring, and $f: R\mapsto R~$ an additive map such that $$f(xy)=f(x)f(y)+f(x)y+xf(y)\quad \forall~x,y\in R $$ It is clear that if we take any endomorphism $g$ of R then $g-id_R$ verify ...
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1answer
49 views

Let $M = K + L$ and let $f: M \to N$ be an epimorphism. Prove that $N = f(K) \oplus f(L)$ if $K \cap L = \operatorname{Ker}f$

Let $M = K + L$ and let $f: M \to N$ be an epimorphism. Prove that $N = f(K) \oplus f(L)$ if $K \cap L = \operatorname{Ker}f$ I think we should start with take an element from $K + L$ and then we can ...
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16 views

How to count the number of selected element in each slot within the packed ciphertext

Given an encrypted ciphertext (n slots, packed n elements into a single ciphertext), such as $ct=\{(2,0,1,2),(3,2,1,3),(3,4,0,4),(5,1,4,2)\}$. Formally, $n$ slots can be expressed as $m$ blocks, each ...
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1answer
21 views

Kernel of ring homomorphism from a polynomial ring over field to a field is maximal ideal or zero ideal

Question If $E,F$ are fields and $\beta:F[x]\rightarrow E$ a homomorphism of rings. Show that the kernel of $\beta$ is a maximal ideal or a zero ideal. I just wonder where does the zero ideal case ...
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1answer
51 views

Why must a ring homomorphism from $\mathbb Q[\sqrt n] \to R$ be the identity mapping?

Let $\mathbb Q[\sqrt d]:= \{a+b\sqrt d \, |\, a,b \in \mathbb Q\}$, and $d\in \mathbb N$ with the condition that $d$ is not a square number. I have shown that this is a subring of $\mathbb R$ and that ...

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