0
votes
0answers
17 views

What is $M\otimes_\mathbf{Z}(N/P)\cong ?$

Let $A=\mathbf{Z}[G]$ where $G$ finite, and let $M,N,P$ be $A$-mods. Is there any particularly nice relationship for $M\otimes_\mathbf{Z}(N/P)$?
0
votes
1answer
10 views

Isomorphism of a torsion product and a quotioten of torsion product.

I have the following problem: If $A'$ a submodule of the right $R$-module $A$ and $B'$ a submodule of the left $R$-module $B$, then $A/A' \otimes B/B' \cong (A\otimes B)/C$ where $C$ is the ...
2
votes
1answer
28 views

Tensor product and its homomorphisms.

Given $f:A_R\rightarrow A'_R$, $g:B_R\rightarrow B'_R$ R-module homomorphism we can define $f\otimes g: A\otimes_R B\rightarrow A'\otimes_R B'$ such that $(f\otimes g)(a\otimes b)=f(a)\otimes g(b)$ ...
1
vote
0answers
26 views

Why does $\rho_{\mathbf{Z}^t\otimes P}=\rho_R$ imply the isomorphsim of $\mathbf{Z}^t\otimes P\cong R$?

so I have happened upon a thesis regarding the calculations of various $\mathbf{Z}D_6$ modules and their isomorphisms and came across a technique which is bothering me. Let me give an example. Let ...
1
vote
0answers
53 views

Tensor product of free modules over free algebra

Suppose $M$ and $N$ are modules over a (commutative, unital) ring $S$. Let $R$ be a subring of $S$ such that $S,M$ and $N$ are all free, finitely generated modules over $R$. Question: Under what ...
1
vote
0answers
59 views

Surjectivity implies injectivity of finitely generated modules, localization?

The following problem is canonical: Suppose $A$ is a commutative unitary ring, and $M$ is a finitely generated module over $A$. If an endomorphism $f\colon M\to M$ is surjective, then it's also ...
0
votes
0answers
40 views

Exercise 7.10 Atiyah, $M[x] $ is a noetherian $A[x] $-module [duplicate]

The exercise is: Let $M$ be a noetherian $A$-module. Then $M[x] $ is a noetherian $A[x] $ module. The action of $A[x] $ on $M[x] $ is the obvious one. In a previous exercise it was shown that ...
3
votes
0answers
29 views

Tensor product of $A_n$ modules/ localisation at ring of differentials

I'm working through Coutinho's "A Primer of Algebraic D-Modules" and I've gotten stuck on the following question: Let $p \in K[x_1, \ldots ,x_n]$ be non-zero, and let $A_n$ be the Weyl Algebra. Show ...
4
votes
2answers
125 views

Proving that tensoring a projective module with a flat module gives a projective module?

If $P$ is a projective module and $M$ is a flat module, both over some commutative ring $R$, then how do you prove that $M\otimes_R P$ is flat? I tried using the fact that $P$ is the direct ...
0
votes
1answer
26 views

Tensor product and free $R$-module construction

I am trying to understand an exposition on Tensor products by Keith Conrad. In the proof of Theorem 3.2 on page 7 it considers the free $R$-module on the set $M \times N$: $$F_R(M \times N) = ...
2
votes
2answers
47 views

Relation between $\operatorname{Coker}(f)$ and $\operatorname{Coker}(f \otimes 1_P)$

Let $M,N,P$ be $R$-modules ($R$ commutative ring with $1$) and let $f:M\to N$ be a $R$-module homormorphism. Let tensor the homomorphism to get $ f \otimes 1_P : M \otimes P \to N \otimes P $. I ...
1
vote
1answer
65 views

Completion of a torsion-free module

Let $R$ be a Dedekind domain, $K$ its field of fractions, $P$ a non-zero prime ideal of $R$. Let $\hat R_P$ be the completion of $R$ wrt to the valuation $v_P$ induced by $P$ and let $L$ be a ...
1
vote
1answer
43 views

Localization of a torsion-free module

I have a Dedekind domain $R$ with field of fractions $K$ and a non-zero prime ideal of $P$ of $R$. Let $L$ be a torsion-free $R$-module. How can I show that $R_P\otimes_R L$ is torsion-free as an ...
2
votes
1answer
39 views

When is $M \times N$ contained in$ M \otimes_{R} N$.

Let $R$ be a ring with 1. $M$ is a right R-module and $N$ is a left R-module. A tensor product of M and N comes with a map from $M \times N \to M\otimes N$ which is actually a composition of maps ...
0
votes
1answer
44 views

tensor product of R-algebra and f.g module [closed]

$R$ is a commutative noetherian ring. If $S$ is an $R$-algebra, and $M$ a finitely generated $R$-module, is $M\otimes_RS$ finitely generated $S$-module? I only need a hint. Thanks!
2
votes
0answers
52 views

$M_{\mathfrak{p}} \otimes_{R_{\mathfrak{p}}} N_{\mathfrak{p}} = 0$ implies $M_{\mathfrak{p}} = 0$ or $N_{\mathfrak{p}} = 0$ [duplicate]

Studying commutative algebra I've found this statement: If $M$ and $N$ are finitely generated $R$-modules, with $R$ a commutative ring, and $\mathfrak{p} \subset R $ is a prime ideal, then ...
4
votes
1answer
58 views

$2 \otimes_{R} 2 + x \otimes_{R} x$ is not a simple tensor in $I \otimes_R I$

This question is related to Properties of the element $2 \otimes_{R} x - x \otimes_{R} 2$. Another exercise of Dummit-Foote is to show that Let $I = (2, x)$ be the ideal generated by $2$ and $x$ ...
1
vote
1answer
59 views

$\biggl( \prod_{i=1}^{+ \infty} \mathbb{Z}/2^i \mathbb{Z} \biggr)\otimes_{\mathbb{Z}}\mathbb{Q} \neq 0$ without using $p$-adic integers

I'm doing this exercise from Dummit-Foote: Show that tensor products do not commute with direct products in general. [Consider the extension of scalars from $\mathbb{Z}$ to $\mathbb{Q}$ of the direct ...
3
votes
2answers
42 views

$M \otimes_R K$ is uniquely divisible as $R$-module

A module $M$ over a domain $R$ is called uniquely divisible if for every $r \in R \setminus\lbrace 0 \rbrace$ the endomorphism $$\phi_r : M \to M$$ $$m \mapsto r \cdot m$$ is a bijection. Let $K$ be ...
6
votes
0answers
76 views

When does the tensor product consist of elementary tensors only?

The question is: Assume that $R$ is a (commutative) ring. Under what conditions on $R$-modules $M,N$ does the tensor product $M\otimes_RN$ consist of elementary tensors only? That is, every ...
0
votes
2answers
47 views

Tensor product of a vector space and a field

Let $F$ be a field and $V$ a vector space of finite dimension $n$ over $F$. Let $\overline{F}$ be the algebraic closure of $F$. and let $\overline{V}=\overline{F}\otimes_F V$ the tensor product over ...
1
vote
1answer
147 views

Can one prove the existence of tensor product without explicitly constructing it? [duplicate]

R is a ring with 1. We construct tensor product $M \otimes N$ of right R-module $M$ and left R-module $N$ to basically be able to state its universal property that any R-bilinear map from $M\times ...
3
votes
0answers
53 views

When is $N\otimes_A B \to N$ an isomorphism?

Let $A, B$ be commutative (unital) rings and $f\colon A \to B$ an $A$-algebra. There then exists a canonical functor $f_*\colon \mathbf{Mod}_B \to \mathbf{Mod}_A$ such that, for every morphism of ...
2
votes
1answer
35 views

Is this element necessarily non-zero in the tensor product?

Suppose $R$ is an integral domain and $M$, $N$ are torsion-free $R$-modules. If $m$ and $n$ are nonzero elements of $M$ and $N$ respectively then does it follow that $m \otimes n \neq 0$ ? If either ...
2
votes
1answer
57 views

$M \otimes_Z N =0$ if and only if $M \otimes_R N =0$?

If $R$ is a commutative ring, and $M$ and $N$ are $R$-modules, is $M \otimes_Z N =0$ if and only if $M \otimes_R N =0$? The forward direction seems clear. The backward direction seems like it should ...
1
vote
2answers
56 views

Does tensoring with a module preserve this injection even if the module is not flat?

Suppose $R$ is a domain and let $Q$ be its field of fractions. Then $ 0 \to R \to Q $ is exact. Now suppose that $M$ is a torsion free $R$ module. Is it necessary that $ 0 \to M \to Q \otimes M $ ...
0
votes
1answer
31 views

Constructing nontrivial $\mathbb{Z}$-bilinear map from $\mathbb{R} \times (\mathbb{R} / \mathbb{Z})$

I'm trying to show $\mathbb{R} \otimes_\mathbb{Z} (\mathbb{R} / \mathbb{Z})$ is nontrivial, where my tensor product is defined using the universal property. This problem can be easily reduced ...
1
vote
0answers
48 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
61 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
69 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
28 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?
17
votes
2answers
332 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
1answer
76 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
43 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
86 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
52 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
37 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
64 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
29 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
42 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
29 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
39 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
85 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
106 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
107 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
41 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
90 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
50 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
70 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
83 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 ...