1
vote
0answers
49 views

Describe the automorphisms of this $\mathbf{Z}$-module

This follows up on what I thought was a good question which has now been deleted, asking about the automorphisms of the multiplicative group $\mathbf{Q}^{*}$. Is there a relatively simple ...
3
votes
2answers
28 views

group of module homomorphisms is a module

I am trying to solve the following problem: Let $M$ and $N$ be two left $A$-modules. Prove that $Hom_A(M,N)$ has a left $Z(A)$-module structure with: $(a.f)(m)=a.f(m)$. Show $Hom_A(A,N) \cong N$ as ...
0
votes
0answers
30 views

$\mathbb R[X,Y,Z]/\langle Y-X^5,Z-Y^5\rangle \cong \mathbb R[X]$

I am trying to show that the rings $\mathbb R[X,Y,Z]/\langle Y-X^5,Z-Y^5\rangle$ and $\mathbb R[X]$ are isomorphic. I am pretty lost on how to do this. I suppose that the idea is to exhibit an ...
0
votes
1answer
25 views

Ring morphisms, group ring problem

Prove that if $A$ is a ring and $G$ is a group, then the map $$Hom(\mathbb Z[G],A) \to Hom(G,\mathcal U(A))$$ which sends $f \rightarrow f|G$ is a bijection. First of all, I am having some problem ...
0
votes
3answers
40 views

Statements about groups, monomorphism and epimorphism

Problem Let $f:G \to G'$ be a monomorphism. Determine if the following statements are true or false: (i) $G'$ is noncommutative implies $G$ is noncommutative. (ii) $G'$ is cyclic implies $G$ is ...
2
votes
1answer
39 views

Ring homomorphism $f: \mathbb C \to \mathbb C$ such that $f(\mathbb R) \subset \mathbb R$

Exercise Find all the ring homomorphisms $f: \mathbb C \to \mathbb C$ such that $f(\mathbb R) \subset \mathbb R$ My attempt at a solution In a previous problem I've already proved that the only ...
0
votes
1answer
20 views

Statements about ring homomorphisms and division rings

Problem Decide whether the following statements are false or not. 1) If $A$ is a commutative ring such that every ring homomorphism different from the null morphism $\phi:A \to A'$ is injective, ...
0
votes
1answer
33 views

How to show $\alpha=i_L$ and other equalities without using isomorphism?

This is a question of Commutative Algebra.It was given in my class.I was able to solve it to some extend.But I have some doubts. Please help me. Thnx in advance. $A$ is a commutative ring with ...
0
votes
1answer
25 views

Abstract Monomorphism 3 part Question

I have been working on this problem for an hour now and gotten nowhere: Let $G$ be any group and $A(G)$ the set of all 1-1 mappings of $G$, as a set, onto itself. Define $L_a : G \rightarrow G$ by ...
1
vote
3answers
82 views

Characterizing kernel of ring morphism

Let $K$ be a field and define a ring morphism $$\psi: K[x_1,x_2, \dots , x_n, y_1, y_2, \dots , y_n] \rightarrow K(x_1,x_2, \dots , x_n)$$ by $\psi(x_i) =x_i$ and $\psi(y_i) =\frac{1}{x_i}$. I ...
6
votes
2answers
275 views

Linear algebra - Memorising proper definitions of homomorphism types

I am reading a book about linear algebra. On the basis of this book, I worked out the terminology below. Problem: To me, it looks like Wikipedia defines homomorphism differently. Apart from that: Do ...
3
votes
3answers
78 views

An epimorphism from $\mathbb Z⊕\mathbb Z⊕\cdots$ to $\mathbb Q$

I want an explicit example of an epimorphism from $\mathbb Z⊕\mathbb Z⊕\cdots$ to $\mathbb Q$. Thanks.
1
vote
2answers
41 views

Normal subgroups, direct product and monomorphism problem

Let $G$ be a group and let$H,K$ be normal subgroups of $G$. Let $\pi_H,\pi_K$ be the projections on $H$ and $K$ respectively. Show that the map $$f:G/(H \cap K) \to G/H \times G/K$$ defined as ...
0
votes
0answers
34 views

Extending a morphism to a finite algebraic field extension [duplicate]

I am trying to understand the proof of Theorem 5.21 in Introduction to Commutative Algebra, and am stuck on the portion underlined in red (note that $$\Sigma := \{(A,f) \mid A \text{ is a subring of ...
1
vote
1answer
44 views

How do I show that an endomorphism is self-adjoint if and only if $\langle u, Tu \rangle \in \mathbb{R}$ for all $u \in \mathbb{V}$

Let $$(V,\langle \cdot , \cdot \rangle)$$ be a complex vector space. Let $T \in \mathcal{L}(V)$ be an endomorphism. Now I want to show, that $T \in \mathcal{L}(V)$ is self-adjoint if and only if ...
2
votes
2answers
51 views

Homomorphism $f: \mathbb{C}^{*}\rightarrow \mathbb{R}^{+}$. Prove that kernel of f is infinite group.

First of all we need to prove that $\ker(f)$ is group by proving: That $\ker(f)$ contains $e\in\mathbb{C}^*$, That $\ker(f)$ is closed under multiplication for every $a,b \in \ker(f)$ That ...
0
votes
0answers
24 views

Endomorphisms of Groups - Book Recommendation

Which books dealing with group theory have considerable material on endomorphisms? The books I have seen usually have something on homomorphisms, isomorphisms, and automorphisms, but very little on ...
2
votes
3answers
148 views

Group Homomorphism Questions (my attempts shown)

(a) Let $p$ be a prime. Determine the number of homomorphisms from $\Bbb Z_p \oplus \Bbb Z_p$ into $\Bbb Z_p$. Attempt: Suppose $\Psi:Z_p \oplus Z_p \rightarrow Z_p$ is an into homomorphism. ...
2
votes
1answer
35 views

Isomophism between rings an two right ideals

Let I, J two right ideals of a ring R such that I+J =R. Show thath the direct sum of I and J is isomorphic to the direct sum of R and the intersection of I and J. Can anyone please give me at least ...
0
votes
2answers
28 views

Factor ring induced by the ideal generated by x(x-1)(x-2)

Consider the ring $R:=\mathbb C[x]/I$, where $I$ is the ideal in $\mathbb C[x]$ generated by $x(x-1)(x-2)$. Show that the evaluation map $\mathbb C[x]\to \mathbb C,\ p(x)\mapsto p(\alpha)$, for ...
2
votes
1answer
57 views

Does an injective homomorphism always exists from $G$ into $GL_n(R)$ where order of $G$ is $n$?

We have a group $G$ of finite order $n$. Does a one to one homomorphism always exist from $G$ to general linear group?
2
votes
1answer
112 views

How to construct a nonzero homomorphism from a module to a proper submodule?

Let $M$ be a finitely generated module over a commutative ring and $N$ be a non zero proper submodule of $M$. Then is it always possible to have a non zero homomorphism $f$ from $M$ to $N$?
1
vote
3answers
106 views

Automorphism groups of isomorphic groups are isomorphic

Say $G \cong H$ are isomorphic groups. Show $Aut(G) \cong Aut(H)$ I just made this up so I'm not sure if actually $Aut(G) \cong Aut(H)$ is true but I'm $99.9\%$ sure this should be true I'm having ...
5
votes
1answer
79 views

$Gal(\mathbb{Q}(\sqrt 2 + \sqrt 3)/\mathbb{Q})$

A basis for $\mathbb{Q}(\sqrt 2 + \sqrt 3)$ over $\mathbb{Q}$ is $\{1,\sqrt 2 , \sqrt 3 , \sqrt 6 \}$ The roots of $x^2 -2$ are $\pm \sqrt 2$ and the roots of $x^2 -3$ are $\pm \sqrt 3$ so to find ...
0
votes
3answers
48 views

Problem with understanding homomorphism

Let $G$ be the group of all matrices of the form $\begin{pmatrix} 1 & a & b \\ 0 & 1 & c \\ 0 & 0 & 1 \\ \end{pmatrix}$, $a,b,c \in \mathbb R$, ...
1
vote
2answers
90 views

How do I show a mapping is a homomorphism?

I don't want to make this question too broad, or non-specific. I'll will discuss a simple situation so we can all share a common context, but my question is less about this particular group, and more ...
4
votes
2answers
238 views

homomorphism $f: \mathbb{C}^* \rightarrow \mathbb{R}^*$ with multiplicative groups, prove that kernel of $f$ is infinite.

Let $f: \mathbb{C}^* \rightarrow \mathbb{R}^*$ be a homomorphism of the multiplicative group of complex numbers to the multiplicative group of real numbers. I need to show that the kernel of $f$ must ...
2
votes
2answers
189 views

Evaluation morphisms of formal power series and nilpotent elements

Given a commutative ring $A$, and a finitely presented (associative) $A$-algebra $B$, show that a morphism of $A$-algebras $A[[x]] \longrightarrow B$ is given by evaluation at an nilpotent element $ ...
2
votes
2answers
107 views

How Many Homomorphisms $\Bbb{Z}_4 \to \Bbb{Z}_8 \times \Bbb{Z}_{12} \times \Bbb{Z}_{15}$?

I know that the number of homomorphisms between $Z_n$ and $Z_m$ is $\gcd(m,n)$. However, I don't know what to do with these two questions: How many different homomorphisms exist: $\Bbb{Z}_4 \to ...
0
votes
1answer
78 views

Two isomorphism questions

Let G = (C - {0}, mult.), and let U be the subgroup U = {x+yi such that x^2 + y^2 = 1}. Use the Fundamental Theorem to show that: a) G/U is isomorphic to (R>0, mult.) b) G/R>0 is isomorphic to U. ...
0
votes
2answers
119 views

How to show there is only one homomorphism between $\mathbb Z_{25}$ and $S_4$?

Show that there is only one homomorphism from $\mathbb Z_{25}$ to $S_4$. How do i approach this question? If anyone could give me a hint or some guidelines that would be great.
0
votes
2answers
67 views

How is this map a well-defined homomorphism?

If $f: R \rightarrow S$ is a homomorphism of rings with kernel $K$, and $I$ is an ideal in $R$ such that $I \subset K$. The hypothesis is that the map $\overline{f}: R/I \rightarrow S$ given by ...
1
vote
3answers
102 views

No isomorphism between $(\mathbb R,+)$ and $(\mathbb R^*, \times)$

My goal is to disprove the existence of an isomorphism between $(\mathbb R,+)$ and $(\mathbb R^*, \times)$. I proceeded by contradiction. Suppose $f$ is such a map. Then $$f(0-0)=f(0)f(-0)=-f(0)^2$$ ...
1
vote
3answers
141 views

Question about automorphisms

Let G be a finite abelian group and let n be a positive integer relatively prime to |G|. a. Show that the mapping ϕ(x)=x^n is an automorphism of G. b. Show that every x∈G has an nth root, i.e., for ...
5
votes
2answers
86 views

Differences between nilpotent and pointwise nilpotent endomorphisms.

Consider an endomorphism of a module $f:M\rightarrow M$. We have that $f$ is pointwise nilpotent if $\forall x\in M,\ \exists n,\ n\in \mathbb N$ such that $f^{n}(x)=0$. I already know that the ...
3
votes
1answer
79 views

Morphisms, Splitting Fields, and Primitive Cube Roots

Let $\alpha=\sqrt[3]2$ and $\omega=e^{2\pi i/3}$, and let $K=\mathbb{Q}[\alpha,\omega]$ be the splitting field of $f(x)=x^3-2$ over $\mathbb{Q}.$ Determine all morphisms $K\to K$. Ok, so I have ...
0
votes
2answers
82 views

$k$-algebra homomorphisms

I would like to ask if the following is true: Let $A$ be a $k$-algebra where $k$ is any field. If we have a $k$-algebra homomorphism $f:A\rightarrow k$, does it follow that $\ker(f)$ is a maximal ...
2
votes
2answers
72 views

Proving the Frobenius map is an endomorphism

I have prime $p$, and $K$ a field such that $p \cdot 1 = 1+1+\cdots+1 = 0$. I am asked to prove that $F: K \rightarrow K$, $a \mapsto a^p$ is a ring homomorphism. I can prove this for ...
0
votes
1answer
44 views

Say $K=\mathbb Q(2^{1/3})$. Determine all endomorphisms of $K$.

Say $K=\mathbb Q(2^{1/3})$. Determine all endomorphisms of $K$, and justify your answer. Hint: Say $f(x)= x^3-2$. How many roots of $f$ are in $K$? For this I know $x^3-2$ has 1 real root, ...
0
votes
1answer
163 views

Isomorphism of R-modules

Does somebody has an example where the left $R$-modules $R^m$ and $R^n$ are isomorphic for all positive integers $m$, and $n$?
3
votes
3answers
191 views

Isomorphisms and the Fundamental Homomorphism Theorem

Let $$ R=\left\{ \begin{bmatrix} a & b \\ 0 & a \end{bmatrix} : a,b∈ℝ\right\}⊂M_2(ℝ) $$ and $$ I=\left\{ \begin{bmatrix} 0 & b \\ 0 & 0 \end{bmatrix}: b∈ℝ\right\}. $$ Identify the ...
3
votes
1answer
151 views

Automorphism of $\mathbb{Z}\rtimes\mathbb{Z}$

I'm looking for a description of $\mathrm{Aut}(\mathbb{Z}\rtimes\mathbb{Z})$. I've tried an unsuccessfully combinatorical approac, does anymore have some hints? Thank you.
1
vote
2answers
172 views

Is the ring $\mathbb{Z}_5[x]$ isomorphic to the ring of polynomial functions from $\mathbb{Z}_5$ to $\mathbb{Z}_5$?

Is the ring $\mathbb{Z}_5[x]$ isomorphic to the ring of polynomial functions from $\mathbb{Z}_5$ to $\mathbb{Z}_5$? If not, what is a good counterexample? If yes, how can we prove that there's a ...
30
votes
4answers
793 views

$\mathbb C[X]/(X^2)$ is isomorphic to $\mathbb R[Y]/((Y^2+1)^2)$

This question led me to the following: Prove that $\mathbb C[X]/(X^2)$ is isomorphic to $\mathbb R[Y]/((Y^2+1)^2)$.
3
votes
2answers
427 views

Prove the third isomorphism theorem

I'm trying to prove the third Isomorphism theorem as stated below Theorem. Let $G$ be a group, $K$ and $N$ are normal subgroups of $G$ with $K⊆N$. Then $$(G⁄K)⁄(N⁄K)≅G⁄N.$$ I look up for some ...
0
votes
3answers
94 views

Let $p$ be a prime. Determine the number of homomorphisms from $\mathbb{Z}_p \oplus \mathbb{Z}_p$ into $\mathbb{Z}_p$.

Let $p$ be a prime. Determine the number of homomorphisms from $\mathbb{Z}_p \oplus \mathbb{Z}_p$ into $\mathbb{Z}_p$. How can I able to solve this problem?can anyone help me please.
6
votes
3answers
151 views

Aut $\mathbb Z_p\simeq \mathbb Z_{p-1}$

I'm trying to prove that Aut $\mathbb Z_p\simeq \mathbb Z_{p-1}$ (p prime). I know that Aut $\mathbb Z_p$ has $p-1$ elements because $\mathbb Z_p$ has $p-1$ possiblities of generators, so intuitively ...
1
vote
3answers
153 views

Ring homomorphism question.

If $R$ is a ring, show that there is exactly one ring homomorphism $\phi: \mathbb{Z} \to R$. I can't grasp the idea that there can be only one ring homomorphism. Aren't there many (at least more than ...
8
votes
1answer
85 views

Prove that $Tx=x^{-1}~\forall~x\in G.$ [duplicate]

Let $G$ be a finite group and suppose the automorphism $T$ sends more than $\dfrac{3}{4}th$ of the elements of $G$ onto their inverses. Prove that $Tx=x^{-1}~\forall~x\in G.$
3
votes
2answers
666 views

If $\phi:G\to\bar{G}$ is an isomorphism and if $H$ is a normal subgroup of $G$, then $\phi(H)$ is a normal subgroup of $\bar{G}$.

If $\phi:G\to\bar{G}$ is an isomorphism and if $H$ is a normal subgroup of $G$, then $\phi(H)$ is a normal subgroup of $\bar{G}$. I am struggling with getting started with the problem. I know ...