# Tagged Questions

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

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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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### Counterexample: Homology with Local Coefficients versus Homology with Module Coefficients

Definitions Let $X$ be a nice space with universal covering $\widetilde{X}$. Homology $H_*(X,R)$ with Ring Coefficients $R$ is the homology of \begin{align*} C_n(X;R) :=&\; \text{ free left $R$-...
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### Is it $\prod_{i \in \Delta } (S^{-1}A_{i}) \cong S^{-1} \prod_{i \in \Delta }A_{i}$ true? [closed]

Suppose that $A$ is a ring and $S$ is a multiplicative closed subset of $A$. I wonder that do we have $\prod_{i \in \Delta } (S^{-1}A_{i}) \cong S^{-1} \prod_{i \in \Delta }A_{i}$? Can you give me ...
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### Why does this ring have rank $k!$?

Let $R$ be any ring free of rank $k$ over $\mathbb{Z}$ having non-zero discriminant. Let $R^{\otimes k} = R \otimes_{\mathbb{Z}} \cdots \otimes_{\mathbb{Z}} R$. Then $R^{\otimes k}$ is a ring of rank ...
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### Prove that $R \otimes_R M \cong M$

Let $R$ be a commutative unital ring and $M$ an $R$-module. I'm trying to prove $R \otimes_R M \cong M$ but I'm stuck. If $(R \otimes M, b)$ is the tensor product then I thought I could construct an ...
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### Showing that if $R$ is a commutative ring and $M$ an $R$-module, then $M \otimes_R (R/\mathfrak m) \cong M / \mathfrak m M$.

Let $R$ be a local ring, and let $\mathfrak m$ be the maximal ideal of $R$. Let $M$ be an $R$-module. I understand that $M \otimes_R (R / \mathfrak m)$ is isomorphic to $M / \mathfrak m M$, but I ...
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### How to prove that $R/I \otimes_R M \cong M / IM$ [duplicate]

Possible Duplicate: Showing that if $R$ is local and $M$ an $R$-module, then $M \otimes_R (R/\mathfrak m) \cong M / \mathfrak m M$. In one of the answers to one of my previous questions the ...
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### What is the simplest example of the tame representation type?

What is the simplest example of the tame representation type? I tried to find simple example could help me to understand the tame representation type. I know the definition of tame is like: A ...
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### Semisimple objects in abelian categories

Let $\mathcal A$ be any Grothendieck abelian category and $0 \neq M \in \cal A$ an object. It is true that $M$ admits a simple subquotient? It is certainly true for $\mathcal A=R-Mod$ since $M$ ...
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### $f:A\to B$ flat homomorphism of rings, $Q$ is a prime ideal of $B,$ and $P=Q^c.$ Why is $B_Q$ a local ring of $B_P?$

I have a question about Exercise 2.18 on Introduction to Commutative Algebra by Atiyah and MacDonald. Let $f:A\to B$ be a flat homomorphism of rings, let $Q$ be a prime ideal of $B$ and let $P=Q^c,$ ...
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### Sum of free submodules of a module over a PID

It's possible to produce an example of an integral domain $R$ and a free $R$-module $M$ with free submodules $L, L'$ such that $L+L'$ is not free. We can take $R=M=K[x,y]$ , $L=\left<x\right>$ , ...
Let $K$ be a commutative ring. Is it true that $T(V \oplus W) \cong T(V) \otimes_K T(W)$ as $K$-algebras for any $K$-modules $V$ and $W$? The reason I ask is that I have heard it mentioned (I don't ...
The Poincaré–Birkhoff-Witt (PBW) theorem implies that if $K$ is a commutative ring and the Lie $K$-algebra $\mathfrak{g}$ is a free $K$-module, then the unit \$\eta_{\mathfrak{g}}: \mathfrak{g} \to U (\...