# Tagged Questions

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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### Isomorphism between the space of linear operator and matrices for finite dimensional spaces

Prove that $\operatorname{Lin}(U,V)$ is isomorphic to the space of $m$ by $n$ matrices, where $\dim(U)=n$ and $\dim(V)=m$. Thanks so much for your enlightment.
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### Problem on abelian group

Let $G$ be an abelian group, and $\Phi:G\to \mathbb{R}$ is a function with the following property: $$\forall a,b\in G,~~ |\Phi(a+b)-\Phi(a)-\Phi(b)|<c$$ The problem asks to prove the existence of ...
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### Systematizing graph morphisms

Trying to systematize possible notions of graph morphisms I came about the following classification: A morphism $f$ which sends a graph $G$ to another graph $G'$ is – first of all – ...
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### An example of a monomorphism that is not an equalizer in “Abstract and Concrete Categories — The Joy of Cats”

I am sorry, maybe i am just confused or have not understood some definition, but i do not understand the following remark in Abstract and Concrete Categories -- The Joy of Cats on page 117: ...
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### How to denote an 'atomic' morphism in category?

I want to distinguish between two disjoint classes of morphisms in a category: (1) those morphisms that are composed of other morphisms (other than identities) and could conceivably be factored into a ...
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### Morphisms between quasiprojective varieties preserve irreducibility

Let $X,Y$ be two quasiprojective varieties and $\phi \colon X \to Y$ a surjective morphism. Let $Z \subset Y$ a closed set such that $\phi^{-1}(Z)$ is irreducible. Prove that $Z$ is irreducible. ...
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### What does “homomorphism” require that “morphism” doesn't?

I'm starting to learn category theory, but there's one thing I don't get: all morphisms seem to be homomorphisms; the definition seems to be the same. What's the difference between these two? Can you ...
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### To what extent are morphisms required to be functions?

Just beginning category theory, and I am looking for clarification on the precise nature of morphisms. The most familiar categories, e.g. $\mathbf{Top}$, have morphisms that are functions in the ...
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### If $M$ is either Noetherian or Artinian and $M^{(n)}\cong M^{(m)}$, then $m=n$.

The question is: Prove that if $_RM$ is either artinian or noetherian and if $m,n\in\mathbb{N}$ with $M^{(m)}\cong M^{(n)}$, then $m=n$. Here, $M^{(m)}$ denotes the direct sum of $m$ times $M$. I ...
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### Problem 3.1.2 in Liu — Omission in problem statement?

Exercise 3.1.2 in Liu's Algebraic Geometry and Arithmetic Curves is as follows. Let $f:X\rightarrow Y$ be a morphism of schemes. For any scheme $T$, let $f(T):X(T)\rightarrow Y(T)$ denote the ...
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### Finite fiber of scheme morphism is zero-dimensional?

Let $X$ and $Y$ be locally Noetherian schemes and $f:X\rightarrow Y$ be an étale morphism of finite type. Let $x\in X$ and $y=f(x)$. I would like to know why the fiber $X_y$ is a zero-dimensional ...
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### 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 ...
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### Every injective function is an inclusion (up to a unique bijection)

Let $X$ be a set and let $A$ be a subet of $X$. Let $i:A\longrightarrow X$ be the usual inclusion of $A$ in $X$. Then $i$ is an example of an injective function. I want to show that every ...
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### 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.
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### 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 ...
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### 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 ...
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### Algebra (Not *)-Isomorphisms of von Neumann algebras

Let $A$ and $B$ be any two infinite-dimensional von Neumann algebras, they are operator algebras with operator composition as the multiplication and as infinite dimensional vector spaces they're ...
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### 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 $Tx=x^{-1}~\forall~x\in G.$
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### 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 ...
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### Path Connectedness and continuous bijections

Mathoverflow. Are there any two topological spaces $X$ and $Y$ such that they are path connected and such that there exist continuous bijections $X\rightarrow Y$ and $Y\rightarrow X$, but and yet ...
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### A question on morphisms of fields

Let $A,B$ be two fields. Let $\phi:A\rightarrow B$ and $\psi:B\rightarrow A$ be two morphisms of fields. Can i conclude that $A$ and $B$ are isomorphic fields? My guess is yes, because every morphism ...
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### How I can find the inverse of an isomorphism?

The motivation of this question can be found in Can we extend the map $φ$ to $ℝ^{r}×C(ℚ)^{\text{tors}}→C(ℚ)$ as an isomorphism or not? My question is: How I can find the inverse of the ...
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### Order of kernel of a homomorphism

Let $p,q$ be distinct primes. Prove that the kernel of the map $$f: (\mathbb{Z}/p^k\mathbb{Z})^* \rightarrow (\mathbb{Z}/p^k\mathbb{Z})^*$$ defined by $f(x)=x^q$ has order $\gcd(p-1,q).$ Thank ...
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### Problem of monomorphism of rings [duplicate]

Let $A$ a ring, for each monomorphism $f:A^m \rightarrow A^n$, I don't know how to prove that $m\leq n$. I can't start the problem, I have no idea, help me please.
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### Image of a morphism of varieties

Suppose $A$ and $B$ are two algebraic varieties, and $f:A\to B$ is a morphism of algebraic varieties. I guess it is true that $\text{im}(f)$ is itself an algebraic variety. But how to prove it?
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### If each component of a Cartesian product is homeomorphic to another space, are the Cartesian products homeomorphic

Assume we are given a space $A$ with a metric $d$. Assume $A = A_1 \times A_2 \times A_3 \cdots$, ie. $A$ is a Cartesian product of spaces $A_i$, where $i \in I$. $I$ is countable or countably ...
Consider two morphisms: $f : X \to Y$ and $g : Y \to Z$ , and their composition: $g \circ f : X \to Z$. What is the name given to the role of $Y$ with respect to $g \circ f$? Is there a naming ...