# Tagged Questions

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### Tensor product of two simple modules

Let $k$ be a field of arbitrary characteristic. Suppose that $A$ and $B$ are finite dimensional $k$-algebras. Let $S$ be a finite dimensional simple $A$-module and let $T$ be a finite dimensional ...
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### About the definition of the tensor product of modules

I was reading something about tensor product (balanced product) of modules (over an arbitrary ring $R$), but I cannot realize why we need a left and a right module. Would it be the same with two right ...
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### Tensor product is o?

$K$ generated by $(ar,rb)$ for every $a\in A$, $b\in B$, $r\in R$. $F$ generated by $r(a,b)$ for every $a\in A$, $b\in B$, $r\in R$. $(a,rb)=(ar,b)=r(a,b)$,then $K=F$, $F/K=0$ Apparently I ...
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### Tensoring and retaining projectiveness

Let $A$be a unital associative ring, If $A$ is not a projective $A$-bimodule, however $A\otimes A$ is a projective $A$-bimodule, can we conclude that $A\otimes A \otimes A$ is also projective?
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### If the tensor power $M^{\otimes n} = 0$, is it possible that $M^{\otimes n-1}$ is nonzero?

Let $M$ be a module over a commutative ring $R$. It is possible that $M \otimes M = 0$ if $M$ is nonzero, for example when $R = \mathbb{Z}$ and $M = \mathbb{Q}/ \mathbb{Z}$. What about when higher ...
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### Tensor powers of injective linear maps of free modules

This is a basic question on tensor products of linear maps. Let $R$ be a commutative ring and let $\varphi: M\to N$ be an injective linear map of finitely generated free $R$-modules. Question: Are ...
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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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### Extension and restriction of scalars and their relation to the identity functor

Let $R$ and $S$ be rings with $R \subseteq S$, and let $_S M$ be a left $S$-module. If we restrict our scalars to $R$, we naturally have a left $R$-module structure on $M$, $_R M$, given by ...
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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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### Tensor product of certain algebras

If S is an $R$-algebra of finite dimension, and $A$ is an algebra of infinite dimension, then is $S \otimes S \otimes A \cong S \otimes A$?
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### How to calculate the tensor product?

This question might be stupid for you.I ask because I have no clue about it. I don't really understand what is tensor product,although I know its definition. I have search what is tensor product,so I ...
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### What is the most generic algebraic structure for which we can define a tensor product? [duplicate]

We can define a tensor product of two vector spaces. But vector spaces are themselves modules and we can also define a tensor product of two modules. My question is the following: are modules the ...
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### Tensor of quotients question

Let $R = \mathbb{Z}[x]$ be the 1-variable polynomial ring over the the integers, $q$ and $p$ arbitrary prime numbers, $R_1 = \mathbb{Z}[[x]]/(p)$, and $R_2 = \mathbb{Z}[x]/(q, x)$. Question: what ...
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### Adjoints to cofree modules tensor?

If $M$ is a cofree $R$-module and $A,B$ are arbitrary $R$-modules then, is there a left adjoint to the functor $M\otimes_R -$, i.e. is there an endofunctor $F$ on $_R \mathrm{Mod}$ such that ...
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### Calculating a certain tensor

Consider the $\mathbb{Z}$ module $\mathbb{Z}/n\mathbb{Z}$. What is $\mathbb{Z}/n\mathbb{Z} \otimes_{\mathbb Z} \mathbb{Z}/n\mathbb{Z}$?
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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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### Tensor products and monomorphisms of $A$-modules.

This problem addresses the same question that has been asked in $A^m\hookrightarrow A^n$ implies $m\leq n$ for a ring $A\neq 0$, which is exercise 2.11 of Atiyah and Macdonald's Introduction to ...
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### If for $A$ a commutative nonzero ring $A^m ≅ A^n$ as $A$-modules, then $m = n$

This is the problem I need to solve: Let $A$ be a nonzero ring. Show that if $A^m ≅ A^n$, then $m = n$. The book I got this problem from suggests using the following method to solve it: Let ...
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### Free Module over a tensor product

Suppose that $A$ and $B$ are $\mathbb{K}$-algebras for some field $\mathbb{K}$. Is $B$ free over $A \otimes_{\mathbb{K}} B$ ?
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### Operation on a Tensor Product of Modules

Suppose that $M$ and $M'$ are $\mathbb{K}$-algebras where $\mathbb{K}$ is a field. Now suppose that $N$ is an $M$-module and $N'$ is an $M'$-module, then I can view both of them as ...
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### A Property of Tensor products

So I'm new to tensor products and there's something that's been confusing me ... Suppose that $A$ and $B$ are $R$-modules, I know that $A \otimes_R B$ is an abelian group; and what I understood is ...
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### Tensor product of modules preserve injectiveness and surjectiveness or not?

Let $R$ be a commutative ring with identity and $M$ an $R$-module. If $N_1\longrightarrow N_2$ is injective (resp. surjective), is the induced map $M\otimes_R N_1\longrightarrow M\otimes N_2$ ...
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### Question on Criterion for Flat Modules

I'm reading Categories and Modules with K-Theory in View by A. J. Berrick, M. E. Keating, on Flat Modules and there's one part that I'm not quite sure that I fully grasp it. It's on page 155. ...
### If $x\otimes y=0$, then $x\otimes y=0$ in the tensor product of finitely generated submodules
Let $A$ be a commutative ring, let $E,F$ be $A$-modules. Let $x\in E$ and $y\in F$. If $x\otimes y=0$ in $E\otimes F$, then there exist finitely generated submodules $M\subset E$ containing $x$ and ...
I'm looking for an example of modules $A,B$ such that $A\otimes_{R}B\neq A\otimes_{\mathbb{Z}}B$. I've thought about some tensor products I know but I haven't been able to produce an example. What ...