# 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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### $k$-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 ...
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### 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 ...
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### 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, ...
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### 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 $n$?
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### 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 ...
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### group,subgroup and isomorphism

I study group theory now but I could not understand isomorphisms very well. In the book that I study I have seen that; $\mathbb{Z}_6=\{0,1,2,3,4,5\}$ is given and $H=\{0,2\}$ is a subgroup of the ...
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### 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.
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### Finding homomorphisms and kernels from a given ring R

Give the following rings $R$ and ideals $I$ find a ring $S$ and a homomorphism $f:R \rightarrow S$ with kernel $I$ i) $R=\mathbb{Q} [x], I=(x^{2}-2)R$ ii)$R=\mathbb{Z}[i], I=2R$ (Gaussian Integers) ...
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### number of automorphisms for group in order 169

Let $G$ be a group with order 169. Prove number of automorphisms is at least 143. I thought that 169 is 13 squared so maybe G isomorphic to $Z_{169}$ but I dont have any idea. How can I solve ...
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### Hypercube and dihedral group

Let $G_n$ denote the subgroup of the orthogonal group $O_n$ of elements that send the hypercube to itself, the group of symmetries $C_n$, including the orientation-reversing symmetries. It would ...
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### Homomorphism on the group of isometries

Prove that the map $f: M \rightarrow \{1,r\}$ defined by $t_a \rho_{\theta} \mapsto 1$, $t_a \rho_{\theta}r \mapsto r$ is a homomorphism. M denotes the set of isometries of the plane; r the reflexion ...
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### Can objects repeat in commutative diagrams?

Are objects allowed to repeat in commutative diagrams? This seems to be necessary when representing endomorphisms such as the morphism $f : X \to X$ in the category $\mathbf{Set}$, such as when $f$ is ...
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### Exponentials “commutes” explicitly

i have a question about exponentials in a category $\Bbb{A}$. I have to prove that the following holds: $C^{A\times B}\cong(C^A)^B$. Therefore i have to give two arrows. This can be done on an ...
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### Functor whose values on morphisms are monomorphisms

Is there a name for a functor whose values on morphisms are monomorphisms?
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### A notation for a morphism in a thin category

Consider a thin category with objects $A\leq B$. There exists a unique morphism $A\rightarrow B$. Is there a standard notation for this morphism (given $A$ and $B$)?
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### 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 ...
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### Number of kernals of all $\mathbb{Z}_n$ to $\mathbb{Z}_m$ homomorphisms?

How many subgroups $K \le \mathbb{Z}_n$ are there with $K =\ker(\phi)$ for some homomorphism $\phi\colon\mathbb{Z}_n \rightarrow \mathbb{Z}_m$? Stuck and need a hint. Have so far that there are ...
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### Bijection abstract simplicial complex

Given two compact Hausdorff spaces $X$ and $Y$ and $h \colon X \to Y$ a homeomorphism, how can I prove that $h_{\mathfrak{A}} : N(\mathfrak{A}) \to N(h(\mathfrak{A}))$ is a bijection where ...
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### Inverse of open affine subscheme is affine

This seems ridiculously simple, but it's eluding me. Suppose $f:X\rightarrow Y$ is a morphism of affine schemes. Let $V$ be an open affine subscheme of $Y$. Why is $f^{-1}(V)$ affine? I noted that ...
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### Why is this composition of scheme morphisms proper?

I am learning about proper morphisms from Liu's book. I have a question about the proof of the Lemma 3.17 on page 104. Let $A$ and $B$ be rings and suppose $\operatorname{Spec} B$ is proper over ...
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### $\mathbb C[X]/(X^2)$ 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 $\mathbb R[Y]/((Y^2+1)^2)$.
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### Category with endomorphisms only

How is called a category with endomorphisms only? How is called a subcategory got from an other category by removing all morphisms except of endomorphisms?
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### Does every category have a functor?

Is there any one (or more) categories that doesn't have a functor? Functors go between categories, so is there any category that only has an identity functor but no other functor that maps it to ...
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### Homomorphical Equivalence is NP-complete

Two graphs $G,H$ are homomorphically equivalent if there are exists a homomorphism from $G$ to $H$ and a homomorphism from $H$ to $G$. The task is to prove that this decision problem ...
I have a very common situation, for which I need both: (1) notation; and, if available, (2) a general relative term. Let's say that: there is a functor between categories, $f:C_1\to C_2$, $c_1$ is ...