Tagged Questions

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Tensor products over monoids : Element structure

Let $A$ be a (commutative) monoid. Let $M$ be a right $A$-set and let $N$ be a left $A$-set. Then we can construct the tensor product $M \otimes_A N$, which is a set (of even $A$-set when $A$ is ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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!
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$M_{\mathfrak{p}} \otimes_{R_{\mathfrak{p}}} N_{\mathfrak{p}} = 0$ 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 ...
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$\mathcal{O}_{X_y,x}=\mathcal{O}_{X,x}/\mathfrak{m}_y\mathcal{O}_{X,x}$

In a proof (proof of theorem 4.3.36 in Liu's book) I need the equality $\mathcal{O}_{X_y,x}=\mathcal{O}_{X,x}/\mathfrak{m}_y\mathcal{O}_{X,x}$. The hypothesis of the theorem are the following: $Y$ ...
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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 ...
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Idempotents in $\mathbb{Z}[i] \otimes_{\mathbb{Z}} \mathbb{Z}[i]$

Letting $\mathbb{Z}[i]=\left\{a+bi:a,b \in \mathbb{Z} \right\}$ be the ring of Gaussian integers, how many idempotents are there in $\mathbb{Z}[i] \otimes_{\mathbb{Z}} \mathbb{Z}[i]$? I came ...
Let $R$ be a polynomial ring over $\mathbb{C}$. Let $R_1=R/I$ for some ideal $I \subset R$. Let $M_1, M_2$ be $R_1$-modules. So, they are $R$-modules as well. Is it true that $M_1 \otimes_{R_1} M_2 ... 1answer 73 views Decomposable Tensors over Rings Suppose$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}$, ... 3answers 131 views When does there exist a commutative ring$C$that contains rings$A$and$B$as a subring? The statement I'm trying to prove is the following: Let$A$and$B$be commutative rings, both of characteristic$0$. Then there exists a commutative ring$C$that contains both$A$and$B$as ... 2answers 170 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 ...
Suppose $R' \subset R$, $S'\subset S$ are inclusion of $k$-algebras. Does it hold that $R'\otimes_kS' \rightarrow R \otimes_k S$ is injective ? I know there're counterexamples for modules, but ...