# Tagged Questions

For questions about modules over rings, concerning either their properties in general or regarding specific cases.

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### How to solve ax+k=bx mod c?

How to solve $a*x+k=(b*x) \mod c$ where $k \in Q$ $a , b ,c \in Z$ thanks
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### Question on projective modules, a course in homological algebra

Show that if $0\rightarrow N\xrightarrow{\tau} P\xrightarrow{\epsilon} A\rightarrow 0$ and $0\rightarrow M\xrightarrow{\tau'} Q\xrightarrow{\epsilon'} A\rightarrow 0$ are exact sequences with $P,Q$ ...
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### Modules that allow an infinite exact sequence on them

I'm looking for a characterization of modules that fit the following property: (*) There's an infinite exact sequence with $\phi_i \neq 0 \space \forall i\in I$ $\require{AMScd}$ \begin{CD}\cdots @&...
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### About the definitions of direct product and direct sum of modules.

Direct product of modules can contain infinitely nonzero elements, but direct sum of modules must contain finitely nonzero elements. What's the point of the definitions? Why the definitions are ...
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### If $N$ is finitely generated, then $(L :_{R} N)^{e}= (S^{-1}L :_{S^{-1}R} S^{-1}N)$ [duplicate]

I have question in this lemma. Please help me explain it more. Let $L$, $N$ be submodules of a module $M$ over a commutative ring $R$ and let $S$ be a multiplicatively closed subset of $R$. If $N$ is ...
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### How many solutions of $\mod 63$ : $x^2=1 \pmod7$ and $x^3=1\pmod 9$ [closed]

How many solutions $\mod 63$ , we have for: $$x^2=1 \pmod 7$$ and $$x^3=1 \pmod 9$$ Need to find them also.
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### why the algebra $A = k[x]/(x^n)$ has finite representation type? [closed]
Suppose that $k$ is algebraically closed. Then why the algebra $A = k[x]/(x^n)$ has finite representation type? please clarify the answer.
### Question about $\mathbb{Z}^2 \otimes \text{sym}^2\mathbb{Z}^3$.
I am confused about the set $\mathbb{Z}^2 \otimes \text{sym}^2\mathbb{Z}^3$... Could someone please explain me why this corresponds to the set of pairs of symmetric $3$ by $3$ matrices? Thank you!