For question about morphism between groups, ring, topological spaces, vector space, categories, etcs... Please also use the correspondent tags (e.g. (group-theory), (ring-theory)) in order to precise the involved structure.

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3
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1answer
30 views

Direct sum of simple modules and Schur's Lemma

Suppose $M,N$ are two non-isomorphic simple $R$-modules. For $m,n\geq1$, is it true that $$ \text{Hom}_R(M^{\oplus m},N^{\oplus n})\cong\hat{0}\,? $$ I think it's true by Schur's Lemma. ...
0
votes
1answer
15 views

Two quotient morphisms and universal property

I am reading some notes on group theory and I am having some doubts related to the following: Let $S \lhd G$ and let $\rho:G \to Q, \space \rho': G \to Q'$ be two quotients of $G$ by $S$. Then, by ...
2
votes
0answers
43 views

Is proper morphism from affine scheme affine?

I'm reading Mumford-Oda's lecture notes http://www.math.upenn.edu/~chai/624_08/mumford-oda_chap1-6.pdf. And they use the fact:"Let $f:U \to Y$ be a proper morphism of noetherian schemes and $U$ is ...
1
vote
1answer
33 views

Proving an isomporphism between all real 2x2 matrix under addition and $R \oplus R \oplus R \oplus R$

Here is my current issue: Let $M$ be the group of all real 2x2 matrices under addition. Let $N=R \oplus R \oplus R \oplus R$ be a group under vector addition. Prove the $M$ and $N$ are isomorphic. ...
0
votes
0answers
10 views

Affine space and vector space isomorphisme

Let $X\ \mathbb{K}$-be vector space, ($X,\overrightarrow{X}$) an affine space and $O$ et $O^{'}$ let $f: X_{O} \to X_{O^{'}} P \mapsto f(P)=P+\overrightarrow{OO^{'}}$ $X_{0}$ design The ...
0
votes
2answers
32 views

Endomorphism example of infinite vector space

Give an example of infinite vector space $V$ and it's endomorphism $F: V \rightarrow V$ that : a) $F$ is Monomorphism but not Epimorphism b) $F$ is Epimorphism but not Monomorphism Thanks for help.
0
votes
2answers
25 views

Question about composition of categories

I'm reading the following to get a high-level overview of categories. http://en.wikibooks.org/wiki/Haskell/Category_theory In this, they have this image: and the author makes this claim: So ...
0
votes
0answers
49 views

(Updated): Finding the kernel of a ring morphism

Let $m,n \in \mathbb{Z} \setminus \lbrace 0 \rbrace $, consider $$\varphi: \begin{cases} \mathbb{Z}_{/<m \cdot n >} &\longrightarrow \mathbb{Z}_{/<m>} \times ...
1
vote
1answer
28 views

ring morphism from a group ring to another ring

I've read that if $S$ is a commutative ring, then $Hom_R(R[G],S)=Hom_R(R,S)\times Hom_{Gr}(G,\mathcal U(S))$. I've tried to show this equality but I couldn't. If $\phi: R[G] \to S$ is a ring ...
2
votes
1answer
39 views

Relationships between initial/terminal objects and initial/terminal morphisms (if any) in the same category.

The definition of initial and terminal objects given here http://en.wikipedia.org/wiki/Initial_and_terminal_objects makes sense to me. The definition of initial and terminal morphisms given here ...
1
vote
1answer
41 views

Dimension and morphism with finite fibers

I'm studying the dimension of projective varieties and in the literature I'm reading I have the following statement: "If $f : X → Y$ is a morphism with finite fibers, i. e. such that $f^{−1}(P)$ ...
1
vote
0answers
50 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
29 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
31 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
29 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
42 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 ...
1
vote
1answer
51 views

degree of morphism of schemes

Let $\phi: Y \to X$ be a finite etale morphism of proper smooth connected schemes over a field $K$ and suppose that the induced morphism $\phi: \overline{Y} \to \overline{X}$ has degree $n$, where ...
0
votes
0answers
27 views

Automorphism of $(\mathbb{R}, <)$ that takes a finite set of reals to a set of naturals

I was working on another of Enderton's logic book exercises (specifically, exercise 20 (b) from section 2.2, p. 102), and, as part of the exercise, he asks us to show that, for any finite set of real ...
2
votes
1answer
41 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
21 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 ...
2
votes
0answers
43 views

Group's morphisms

I know that the constant map equal to $1$ and the signature are two group's morphisms from the group of permutations $(\mathcal S_n,\circ)$ to the group $(\Bbb C^*,\times)$. My question is: Can we ...
2
votes
0answers
105 views

Must diagrams be commutative?

Given a category C and a function $ \Theta : Mor(\textbf{C}) \times Mor(\textbf{C}) \longrightarrow Mor(\textbf{Rel}) $ and suppose that the relation, with $(r,s)\in\Theta(u,\bar{u})$ and so forth, ...
1
vote
3answers
92 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
284 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 ...
2
votes
1answer
37 views

Equality of sets of local isomorphisms between relations

I'm still working on the first pages of Poizat's A Course in Model Theory. I'll state the basic definitions again, in order to avoid referring back to an early question: Poizat defines an isomorphism ...
2
votes
0answers
51 views

On the back and forth conditions for a set of partial isomorphisms

I've recently begun reading Poizat's A Course in Model Theory and already in the first pages I had some doubts. One odd (not necessarily bad) thing is that he defines notions such as isomorphism only ...
3
votes
3answers
81 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
45 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
2answers
28 views

Isomorphism between quotient groups

Exercise Let $f:G \to G'$ be an isomorphism and let $H\unlhd G$. If $H'=f(H)$, prove that $G/H \cong G'/H'$. As I've shown that $H'\unlhd G'$, I thought of defining $$\phi(Ha)=H'f(a)$$I was trying ...
3
votes
2answers
46 views

Abelian group and morphism equivalent statement

Exercise Show that the following statements are equivalent: $(i) \space G \space \text{is abelian.}$ $(ii) \space \text{the map f: G} \to \text{G defined as} \space f(x)=x^{-1} \space \text{is a ...
0
votes
0answers
24 views

Graph of a regular function. When is the projection on the first component birational?

Let $X$ be an irreducible variety over a field $k$ and $f$ a regular function on some open subset $U\subseteq X$. Let $F\subseteq X\times \mathbb{A}^1_k$ be the graph of $f$ and suppose that the graph ...
1
vote
0answers
54 views

Epimorphisms in two directions

Let $\mathcal{C}$ be a category. Consider the following statement: (S1) Whenever $A,B$ are objects and $\iota_1:A\to B$ and $\iota_2:B\to A$ are monomorphisms, then there is an isomorphism $\phi:A\to ...
0
votes
0answers
35 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
32 views

Properties of varieties that can be lifted under etale morphisms

Let $X$,$Y$ be varieties over some field $k$ of char $0$ (not necessarily algebraically closed). Suppose we have an etale morphism $f : Y \to X$. My question: for what properties P can we say that ...
2
votes
1answer
61 views

Terminal objects of the category of morphisms

I'm reading Basic Category Theory for Computer Scientists by Benjamin C. Pierce and in exercice 1.4.6, he asks what the terminal objects are in $Set^\to$. Let $C$ be a category. The category ...
3
votes
0answers
29 views

Let $h: A \to B$ be a weak homomorphism. Is h$[A]$ a substructure of $B$?

A little bit more precise: let $\mathfrak{A}$ and $\mathfrak{B}$ be two structures. Define a weak homomorphism as a function $h: \mathfrak{A} \to \mathfrak{B}$ such that the folowing conditions are ...
0
votes
0answers
53 views

if a map is bijective, then it has inverse. is the inverse also bijective?

I asked this question because I want to know if given two groups that are isomorphic will always have the same order or not.
1
vote
0answers
44 views

Riesz homomorphism (Banach-Stone theorem)

Let $X,Y$ be compact Haussdorff spaces, and consider $$J:C(X)\to C(Y) $$ a bounded linear bijection such that $J(f\cdot g) = J(f)J(g)$. I know that if $T:C(X)\to C(Y)$ is a linear bijection, being ...
0
votes
0answers
39 views

Hypergraph notation and hypergraph morphisms

There are two parts to my question. The first part is about notation for hypergraphs. The sconed is about the notion of morphisms for hypergraphs. For the notation part, the context is that I make ...
1
vote
1answer
46 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
53 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 ...
2
votes
1answer
42 views

Canonical isometric isomorphism of $l_{\alpha}^{2}$

Let $\alpha \in \mathbb{R}$ and $l_{\alpha}^{2}$ the vector space of bi-infinite sequences $(x_{n})_{n\in \mathbb{Z}}$ such that $||x||_{\alpha}:=\sum_{n\in\mathbb{Z}} ...
1
vote
1answer
44 views

Left shift operator $L: l^2 \rightarrow l^2$ on the sequence space $l^2$

$$L: l^2 \rightarrow l^2$$ is defined by $$b = (b_1,b_2,...) \mapsto Lb = (b_2,b_3,...)$$. $(Lb)_n = b_{n+1}$ respectively. How can I determine the adjoint endomorphism $L^*$? Kind regards George
0
votes
0answers
25 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 ...
-4
votes
1answer
42 views

Question about nonisomorphic polynomial rings.

Let $n,k > 1$ be positive integers. Define the reduced polynomial rings : $g^k_n = \Bbb R[X_n]/(G^k_n(X_n))$ where $G^k_n$ is a real polynomial of degree $n$ (that keeps the ring reduced). (k is ...
0
votes
2answers
62 views

Inner automorphisms form a normal subgroup of $\operatorname{Aut}(G)$

For an arbitrary group $(G,\cdot)$ let $\operatorname{Aut}(G) = \{f: G \to G \mid f \text{ is an isomorphism}\}$ be the set of all automorphisms of the group $G$. We assume that ...
2
votes
3answers
164 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
41 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 ...