1
vote
0answers
30 views

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 ...
1
vote
1answer
32 views

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 ...
2
votes
2answers
53 views

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 ...
1
vote
0answers
26 views

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?
10
votes
1answer
107 views

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 ...
3
votes
2answers
70 views

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 ...
1
vote
0answers
40 views

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 ...
0
votes
1answer
37 views

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 ...
3
votes
2answers
47 views

$\mathbb{Z}/n\mathbb{Z}$ 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 ...
2
votes
1answer
29 views

Tensor product of certain algebras

If S is an $R$-algebra of finite dimension, and $A$ is an algebra of infinite dimension, then is $S \otimes S \otimes A \cong S \otimes A$?
1
vote
1answer
48 views

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 ...
2
votes
0answers
28 views

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 ...
1
vote
1answer
41 views

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 ...
2
votes
1answer
26 views

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 ...
2
votes
1answer
37 views

Calculating a certain tensor

Consider the $\mathbb{Z}$ module $\mathbb{Z}/n\mathbb{Z}$. What is $\mathbb{Z}/n\mathbb{Z} \otimes_{\mathbb Z} \mathbb{Z}/n\mathbb{Z}$?
3
votes
1answer
73 views

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 ...
1
vote
3answers
84 views

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 ...
1
vote
2answers
96 views

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 ...
-1
votes
1answer
35 views

Free Module over a tensor product

Suppose that $A$ and $B$ are $\mathbb{K}$-algebras for some field $\mathbb{K}$. Is $B$ free over $A \otimes_{\mathbb{K}} B$ ?
2
votes
3answers
84 views

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 ...
1
vote
1answer
47 views

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 ...
0
votes
1answer
60 views

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 $M\otimes_R N_1\longrightarrow M\otimes N_2$ ...
0
votes
3answers
73 views

Question about tensor product of modules, when does $c \otimes r = 0$?

Let $R$ be an integral domain, and let $K$ be its field of fractions. Let $C$ be an $R$-module. Then is $C \cong C \otimes_R R$ injectively contained in $C \otimes_R K$? Define $\phi: C \otimes_R ...
0
votes
0answers
48 views

Show that we have an algebra homomrohpsim

I need to show that we have an algebra homomorphism $\phi: M_n(K)\otimes_KA \simeq M_n(A)$ Where A is a K-algebra and K is some field. I suspect it's really easy but I don't know what to do. Is ...
1
vote
2answers
57 views

Tensor Product problem.

Let $V$ be a vector space over $\Bbb F$, and let $x\not=0,y\not=0 $ be two elements in $V$. I want to show that $x\otimes_{_F} y=y\otimes_{_F} x$ iff $x=ay$ where $a\in \Bbb F$. I know the second ...
1
vote
1answer
39 views

When is a simple tensor equivalent to natural multiplication?

Let $R$ be a commutative integral domain; let $M, N$ be $R$-modules. Let $x_1 \in M$ and let $x_2 \in N$. When does the following equivalence hold? $$x_1 \otimes x_2 = x_1x_2$$ The textbook I'm ...
2
votes
3answers
74 views

$0$-th exterior power, empty product of modules and their tensor product

$\def\finiteprod#1#2{#1_{1}\times#1_{2}\times\dots\times#1_{#2}}$In Lang's algebra he defines the tensor product as a universal object in the category of multilinear maps from $\finiteprod En$ where ...
1
vote
0answers
34 views

Are there other known tensor-like constructions for $R$-modules?

To construct the tensor product of two modules $M,N$, you start with the free group on $M\times N$ and take the quotient of $F(M\times N)$ by the subgroup generated by the elements of the form : $$ ...
0
votes
1answer
40 views

Linearly independent subsets over extension of scalars

Let $k$ be a field, $D_{1},D_{2}$ division rings containing $k$ and $V$ a $k$-vector space of finite dimension and assume also that $V$ is a $D_{1}-D_{2}$ bimodule. Suppose $W$ is a $k$ basis of ...
2
votes
0answers
39 views

Mistake in the proof that a domain is flat as a module over any subring

Where is the mistake in the following argument? I feel that there has to be one, for example by the very existence of this article. Let $R$ be an integral domain and $S \subseteq R$ be a subring ...
2
votes
0answers
61 views

Complete reducibility of tensor product

Let $L$ be a Lie algebra (over a algebraically closed field, not sure if it is relevant). If $V$ and $W$ are two completely reducible $L$-modules, can anyone give a hint on how to show that $V\otimes ...
1
vote
1answer
52 views

Comparison between ordinary products and tensor products

My question is quite simple, I didn't understand this comparison, why the first lines in this example implies the red ones: Thanks
0
votes
1answer
66 views

Decomposable Tensors over Rings

Suppose $R$ is a commutative ring and $M$ is a $R$-module. Then we can define the tensor product $M\otimes_R M$ and more generally the $k$-fold tensor powers $\otimes_R^kM$ for any $k\in\mathbb{N}$, ...
4
votes
2answers
141 views

About the injection $M \hookrightarrow \mathbb Q \otimes_{\mathbb Z} M$.

I want to prove that every abelian group can be embedded in a divisible abelian group. So I tried $M \rightarrow \mathbb Q \otimes_{\mathbb Z} M, m \mapsto 1 \otimes m$. It is obvious that $\mathbb Q ...
2
votes
0answers
88 views

When is the canonical extension of scalars map $M\to S\otimes_RM$ injective?

Let $\alpha:R\to S$ be a map of unital rings, and let $M$ be an $R$-module. We have a canonical map of $R$-modules: ...
2
votes
2answers
84 views

Tensor Product of Submodules

In Keith Conrad's text on Tensor Products, he states on Page 8 of the text that for a fixed ring $R$ and $R$-modules $M$, $M'$, $N$ and $N'$, and maps $f:M\to M'$, $g:N\to N'$ that ...
6
votes
2answers
114 views

Embedding a module into its quotient module

I've got a very basic question on tensor products. Let $R$ be a commutative integral domain, $K$ its quotient field and let $M$ be a $R$-module. Is the map $M \rightarrow K\otimes_R M$ given by ...
0
votes
1answer
66 views

Show mapping involving tensor product is well defined.

Let $R$ be a subring of $S$, let $N$ be a left $R$-module and let $\iota : N \to S \otimes_RN$ be the $R$-module homomorphism defined by $\iota(n) = 1 \otimes n$. Suppose that $L$ is any left ...
2
votes
1answer
66 views

Is the tensor product of 2 free Abelian groups free?

Ok, basically, I think that is true. Let's consider $A$, and $B$ are both free $\mathbb{Z}-$modules, i.e $A = \bigoplus_{i \in I} \mathbb{Z}$, and $B = \bigoplus_{j \in J} \mathbb{Z}$, so we'll have: ...
0
votes
1answer
130 views

Infinite tensor product definition

My question is short and popular: how to define an infinite tensor product of modules over a ring? So, there is an infinite set $I$ and $A$-modules $M_i$. I should understand what $\otimes_{i\in I} ...
21
votes
3answers
383 views

My proof of $I \otimes N \cong IN$ is clearly wrong, but where have I gone wrong?

Ok, I'm reading some thesis of some former students, and come up with this proof, but it doesn't really look good to me. So I guess it should be wrong somewhere. So, here it goes: Let $R$ be a ...
5
votes
1answer
99 views

Length of the tensor product of two modules

I'm trying to prove the following statement: Let $R$ be a commutative ring. If $M$ and $N$ are finite length $R$-modules, then $M \otimes_{R} N$ has finite length, and ...
2
votes
1answer
60 views

proof of $ A\otimes M\cong M $

Let $M$ be an $A$-module. Can someone prove the $A$-module isomorphism: $$A\otimes M \cong M?$$ (By $\otimes$ I mean tensor product.)
5
votes
2answers
134 views

a flatness criterion

I'm having trouble with part (b) of Exercise 10.5.25 from Dummit & Foote (the goal of the problem is to prove that $A$ is a flat $R$-module iff $A\otimes_R I\to A\otimes_R R$ is one-to-one for all ...
0
votes
1answer
64 views

Tensor product of two cyclic modules

Given a commutative ring $A$ (with identity) and two cyclic $A$-modules, $M$ and $N$ with generators $x$ and $y$, respectively. How do you show that $\mathrm{Ann}(x\otimes_A y) = \mathrm{Ann}(x) + ...
3
votes
1answer
94 views

$G$-invariants of tensor products of $G$-modules

Given two $G$-modules $A$ and $B$, are there general criteria which induce $$ (A \otimes B)^G = A^G \otimes B^G $$ where $\sigma(a \otimes b) = \sigma a \otimes \sigma b$? I saw a note stating, that ...
8
votes
1answer
95 views

If a tensor product is free, what can we say about the tensor factors?

Here is what I'd like to prove: Let $R$ be a commutative, noetherian ring, and let $M$ and $N$ be finitely generated $R$-modules. Suppose $M\otimes_RN\cong R$. Does it follow that $M\cong N\cong ...
3
votes
1answer
74 views

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. ...
3
votes
1answer
105 views

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 ...
5
votes
2answers
181 views

Tensor product over different rings are not equal

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 ...