Mathematics Stack Exchange
Mathematics Stack Exchange

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.

A quotient set is a set of equivalence classes of a given set under an equivalence relation. If the original set has some additional structure, the quotient set might inherit it, giving rise to quotient spaces, quotient groups, quotient rings and so forth.

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Is left multiplication well defined on a quotient set of a group?

Let $G$ be a group and $H$ a subgroup, now for every $g \in G$ we define $\sigma_{g}:G/H \rightarrow G/H: xH \mapsto gxH$. Note that we know nothing about the subgroup $H$. My question is whether or not the function $\sigma_{g}$ is even well-defined…
H. de Gracht
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Is it true that $\mathbb{Z}[\zeta_8]/\langle 1+3\sqrt{i}\rangle\cong\mathbb{Z}_{82}$?

I've been thinking a lot about ideals and factor rings, and I came upon the following. First, consider $\mathbb{Z}[\zeta_8]$, which is the cyclotomic ring of integers such that all $z\in\mathbb{Z}[\zeta_8]$ can be written as $\Sigma_{n=0}^3…
AKemats
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Why are monadic categories over $\mathsf{Set}$ cocomplete?

$\newcommand{\set}{\mathsf{Set}}\newcommand{\T}{\mathcal{T}}$Given any monad $(\T,\eta,\mu)$ over $\set$, it is claimed that the Eilenberg-Moore category of algebras $\set^\T$ is cocomplete. More generally, for any well-powered, cocomplete category…
FShrike
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Doubt in a paper involving factor graphs

I am reading from the paper here. Let $X$ be a graph and $G$ a finite group which acts on it (as per the definition in the paper, the group action is by default faithful). Now, we say that an edge $e=\pmatrix{d_1 & d_2}$ (where $d_i$ denote darts)…
user10575
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Doubt on the notation of quotient sets and quotient vector spaces

In pure set theory level, to construct the quotient set we need: $1)$ A set $X$ $2)$ A equivalence relation $\thicksim$ $3)$ With $2)$ and $3)$, we can form the pair $(X, \thicksim)$. $4)$ With $(X, \thicksim)$, we can form a patition on the set $X$…
M.N.Raia
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Process of identifying quotients of polynomial rings

I'm taking an abstract algebra class and we are introducing quotient rings specifically polynomial quotient rings and I'm trying to work out some example problems, but I cannot figure out a general way to approach problems where we're asked to…
Jacob Sanders
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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 initial in $\mathcal{C}$, then $[\mathbf{0}]$ is…
Jordan Barrett
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How general is the fundamental theorem of equivalence relations?

We know that the fundamental theorem of equivalence relations can be stated without reference to sets by using congruence as a conjunction of two propositions as following: If $R$ together with projections $p_0,p_1:R\rightarrow A$ is a congruence…
Jaspreet
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$p: \mathbb{R} \to \mathbb{R}/A$ Prove $p$ is an open map $\iff$ $A$ is open.

$A$ is a subset of $\mathbb{R}$ with more than 1 point and $p: \mathbb{R} \to \mathbb{R}/A$ is the quotient map. Prove that $p$ is an open map $\iff$ $A$ is open. I know if $A$ is open then for each open set $U$ of $\mathbb{R}$, if $U$ does not…
user725757
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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 has only 3 ideals ({0}, J, A)? Let me know if the…
Felipe
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Is there a name for the arc $\mathbb{S}^1 / (x \sim360 - x)$

I was playing with some ideas in a vague way and I have encountered this structure that arises from taking the space of angles $\mathbb{S}^1$ and quotienting it by the relation $(x, 360-x)$ (here $360$ refers to degrees and can be $2\pi$ if you want…
Sidharth Ghoshal
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Definition and construction of a quotient space using a toy example

To make headway into Agebraic Topology I need to precisely understand the definition and construction of the quotient space. Definitions in my textbooks and online have felt handwavy to me, and I don't feel confident in understanding not only the…
Nate
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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 relation $(x,1)\sim(x',1)$ for all $x$, $x'\in…
Sebastián P. Pincheira
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If $A$ has no non-trivial idempotents, then neither does $A/N$

Let $A$ be a commutative, associative, unital, finitely generated algebra over an algebraically closed field $k$. Denote by $N$ the nilradical of $A$, which is the set of all nilpotent elements of $A$ (or equivalently the intersection of all prime…
MyWorld
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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 $z\in\mathbb{Z}[i]$, $z(2+2i)\in\langle 2+2i\rangle$. Since $2+2i+\langle 2+2i\rangle=\langle…
AKemats
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