# Tagged Questions

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### $k[x]/(x^n) \otimes_{k[x]} k[y]/(y^m)$

This is part of an exercise from Eisenbud: $k$ is a field, describe as explicitly as possible a) $k[x]/(x^n) \otimes_{k[x]} k[y]/(y^m)$ b) $k[x] \otimes_{k} k[y]$ Any hint ?
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### Are the generators of the subgroup defining tensor products linearly independent over $\mathbb Z$?

Let $S$ be a (commutative) ring with identity, and let $M$, $N$ be $S$-modules. (I guess if $S$ isn't commutative, I want $M$ to be a right $S$-module an $N$ a left $S$-module.) In the definition of ...
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### Tensor product of quotient rings [duplicate]

$A$ is a commutative ring with unit and $\mathfrak a$, $\mathfrak b$ ideals. I have to show that $$A/\mathfrak a \otimes_{A} A/\mathfrak b \cong A/(\mathfrak{a+b}).$$ Any hint ?
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### Example of $\sum_i a_i\otimes b_i\in M\otimes_AN$ which cannot be written as $a\otimes b$

In the appendix of my commutative algebra text: Note that in general the element of $M\otimes_AN$ is a sum of the form $\sum_i a_i\otimes b_i$ and cannot be necessarily written as $a\otimes b$. ...
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### Examples where $\hat{A}$ is flat as an $A$-algebra (and $A$ is not noetherian)?

Lately I've looked a bit at $f$-adic completions of commutative rings (see for example my last 2 questions), so here's another question concerning the topic: Let $A$ a commutative ring, $f \in A$ not ...
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### How can we show that $\operatorname{Tor}_1^A(\hat{A},G_f/G)=0$?

This question is quite closely related to my last question: Is it always true that $\hat{A} \otimes_A A_f/A \cong \hat{A}_f/ \hat{A}$? Let $A$ a commutative ring, $f \in A$ a non zerodivisor. Let ...
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### Homomorphisms from the base change of a module

Let $A, B$ be commutative rings with one and let $M$ be an $A$-module, $f: A \rightarrow B$ a ring homomorphism. Consider the (right) $B$-module $M \otimes_A B$. What can we say about ...
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### What is $\mathbb{Z}/n\mathbb{Z}\otimes_\mathbb{Z} m\mathbb{Z}$?

I would like to know what $\mathbb{Z}/n\mathbb{Z}\otimes_\mathbb{Z} m\mathbb{Z}$ is isomorphic to, where $n,m\in\mathbb{N}$. Of course there will likely be cases depending on coprimeness and whatnot; ...
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### $\frac{\mathbb{Z}}{m\mathbb{Z}}\otimes_{Z}\frac{\mathbb{Z}}{n\mathbb{Z}} \cong \frac{\mathbb{Z}}{d\mathbb{Z}}$ [duplicate]

I want to prove that $$\frac{\mathbb{Z}}{m\mathbb{Z}}\otimes_{\mathbb{Z}}\frac{\mathbb{Z}}{n\mathbb{Z}} \cong \frac{\mathbb{Z}}{d\mathbb{Z}}$$ where $m ,n \in \mathbb{N}$ and $d = \gcd(m,n)$. Any ...
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### $\mathbb{Z}/n\mathbb{Z}$ is not flat

On the flat module Wikipedia page, it's stated that $\mathbb{Z}/n\mathbb{Z}$ is not flat over $\mathbb{Z}$. But I don't understand their explanation of why. It is said that ...
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### Example of non-noetherian algebras which are tensor products of noetherian algebras

We suppose all rings are commutative with unity. I am looking for examples of a tensor product $B\otimes_A C$ which is not noetherian, where $A$ is a noetherian ring and $B, C$ are noetherian ...
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### Given ring $A$, ideal $I$, and $A$-module $M$, show that $A/I \otimes_A M$ is isomorphic to $M/IM$.

The question is stated as in the title; the hint I am given is to "tensor the exact sequence" $0 \rightarrow I \rightarrow A \rightarrow A/I \rightarrow 0$, which I take to mean using that sequence ...
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### Prove $A_\mathfrak{p} \otimes_A B_\mathfrak{q} = B_\mathfrak{q}$, where $\mathfrak{q}$ prime in $B$

$\require{AMScd}$ Hi, I think I have the answer for this question, but I'm not sure if it's correct. So I would be very glad if someone could have a quick look through it. Let $A$, $B$ be ...
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### Origin of the modern definition of the tensor product

Due to whom is the modern (i.e. via its universal property) definition of the tensor product, and in which article was it communicated?
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