# Questions tagged [quotient-set]

This tag should be used for questions about quotient sets that might not have any other quotient structure or quotient structures that don't have their own tag.

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### Nonunital non commutative ring with 3 ideals...

It is well known that if a (unital commutative) ring A has only three ideals ({0}, J, A), then the quotient A/J is a field. But, what can we conclude about A/J if A is not commutative nor unital but ...
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### Understanding the definition of cones in James Dugundji's.

James Dugundji defines (Topology, chap. VI definition 5.1) a cone in the following manner For any space $X$, the cone $TX$ over $X$ is the quotient space $(X\times I)/R$, where $R$ is the equivalence ...
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1 vote
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### Quotient set of $\mathbb R^2/SO(2)$

I am trying to understand quotient sets with an example of rotations in the plane. I have seen that $\mathbb R^2/SO(2)\cong \mathbb R_{>0}$. As I understand this, the quotient set is the set of ...
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### $R＝\Bbb{R}[x,y]/(x＋y-a,x^2-ax＋b)$ is integral domain when $a^2-4b＜0$. [closed]

Let $a,b∈\Bbb{R}$ and let $R＝\Bbb{R}[x,y]/(x＋y-a,x^2-ax＋b)$. I want to prove that if $a^2-4b＜0$ , then $R$ is integral domain. My try and thougt; This condition means $x^2-ax＋b$ is irreducible, I ...
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### Is this proof correct to show that $\text{Im}(\varphi)$ is countably infinite?

I want to show that the image of the totient function $\varphi: \mathbb{N} \to \mathbb{N}$ is countably infinite. Notice that $\varphi = i \circ b \circ p$ where $p$ is a surjective map, $b$ is a ...
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### What kind of structure is $\mathbb{Z}[i]/\langle 2+2i\rangle$?

Consider the factor ring $R=\mathbb{Z}[i]/\langle 2+2i\rangle$, where $\langle 2+2i\rangle$ is the ideal of the Gaussian integers such that for all $z\in\mathbb{Z}[i]$, $z(2+2i)\in\langle 2+2i\rangle$....
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### Characterizing the product-structure on cosets $O_{h}/C_{2v}$

I consider the set of cosets $Q=G/H$ where $G=O_{h}\simeq S_{4}\times Z_2$ is the full octahedral symmetry group and $H=C_{2v}\simeq Z_2 \times Z_2$, the symmetry group of a body with a $C_2$-axis in ...
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### Finding the quotient set of relation

Let A={1,2,3,4,5,6,7,8,9}. We define (a,b)R(c,d) iff 2|(a-c) ∧ 3|(b-d). R is an equivalence relation on the set AXA. find the quotient set. I got 2 equivalence classes - [(1,3)], [2,4] how do I know ...
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### Products in quotient category

Suppose $\mathcal{C}$ is a category, $\sim$ is a congruence relation on $\mathcal{C}$, and $[-]: \mathcal{C} \to \mathcal{C}/{\sim}$ is the quotient map. I'm able to prove that if $\mathbf{0}$ is ...
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### "Quotient" of category by arbitrary reflexive relation

I went through the definition of the quotient of a category $\mathcal{C}$ by a congruence relation $\sim$ on the morphisms. I was very surprised to find that you can prove $\mathcal{C}/{\sim}$ is a ...
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### Existence of a function from a quotient

On "Introduction to Smooth Manifolds" by John M. Lee, at page 309 we're talking about multilinear algebra. There's a proposition on the Characteristic Property of the Tensor Product Space. ...
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### Proving there exists a unique function decomposition

I am trying to prove the following result. Let $X$ and $Y$ be sets and $\sim$ an equivalence relation on $X$. Let $f: X \to Y$ be a function which preserves $\sim$, and let $\pi$ denote the ...
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1 vote
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### Thinking about the cosets of $\mathbb{Z}[X]/(f(x))$

Let's think about $$\mathbb{Z}[X]/(f(x))$$ First of all, (f(x)) is the ideal generated by $f(x)$ on $\mathbb{Z}[x]$. Let's consider $f(x) = x^d+1$. Then, if we ...
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### How to prove that a function on a quotient set $\mathbb{Z}/n\mathbb{Z}$ is well defined?

I am asked to prove that a function under a quotient set is well-defined. I know that for a function to be well-defined, two mappings or images found in the co-domain/range can't be mapped from the ...
1 vote
116 views

### Special case where $I\subseteq J$ implies $R/J\subseteq R/I$ seems to hold

Consider a ring $R$ and two ideals $I,J$ such that $I\subseteq J$. There is a natural projection $R/I\to R/J$ by the isomorphism theorem. There is, in general, no homomorphism going the other way ...
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1 vote
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### Constructing a quotient set for negative numbers

There's a relation $\rho$ defined as $\forall a,b \in \mathcal{A}$ such that $\mathcal{A} = R - \{0\}$ and $(x,y)\in \rho = x \cdot y > 0$. I proved that this is in fact an equivalent relation and ...
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