# 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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### 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 ...
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### 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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### In $k[x,y]/(xy^2-x)$, $(x)＝(xy)$

Let $k$ be a field In $R＝k[x,y]/(xy^2-x)$. I want to check $(x)＝(xy)$ in $R$. I understand $(x)＝(xy^2)$ but I cannot proceed from here. Thank you in advance.
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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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### Non-equivariant map that may descend to the quotient

Let $X$ and $Y$ be sets equipped with an action of a group $G$ (the same group for both sets). Let now $f:X\to Y$ be a function with the following property: For every $g\in G$, there exists $h\in G$ ...
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### When is the center of the quotient equals 1? [duplicate]

I want to prove or disprove this statement: If $H = Z(G),$ then $Z(G/H) = 1.$ I have seen an example here Between the center of a quotient group and the total center that shows that in some cases that ...
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### Prove equivalence kernel of an injection is the identity relation

Given a function $f$ and an equivalence relation $\sim$ on the domain of $f$, if two equivalent elements with respect to $\sim$ have the same image under $f$, then $\sim$ is called the equivalence ...
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### Tensor products and the elementary notion of the quotient set

I know that the Tensor Product is a well structured mesh of concepts, which relates the notion of Universal Property (UP) (which defines the required algebraic structure) and the existence of such ...
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### Recommended Exercise Lists for Quotient Objects [closed]

I understand the different types of quotient objects, but I still struggle with them - which I think is a result of me neglecting to work on them despite knowing it's a weak area of mine. How I ...
### Why $[0,3]/(1,2)$ is not homeomorphism to $[0,1]$ [closed]
Clearly $[0,3]/(1,2)=[0,1]+ {\rm point} A+[2,3]$. How can I prove those are not homeomorphic
Let $R$ be a ring with $1$. We say that $a,b\in R$ are associate if $a|b$ and $b|a$, i.e. there exist $r,s\in R$ such that $a=rb$ and $b=sa$. This is an equivalence relation on $R$, and I would like ...