Question is : Suppose $I$ is a principal ideal in a domain $R$. Prove that the $R$ module $I\otimes_R I$ is torsion free. Suppose we have $r(m\otimes n)=0$.. Just for simplicity assume that $m\otimes n$ is a simple tensor so we have $m,n\in I$. As $I$ is principal ideal we have $m=pa$ and $n=qa$ for […]

I am trying to show that $\mathbb{C}\otimes_\mathbb{C} \mathbb{C} \cong \mathbb{R}\otimes _\mathbb{R} \mathbb{C}$ as abelian groups. I’ve tried to come up with various maps but gotten nowhere. Thanks for any help

Let $A$ be a commutative, unital ring and let $M_i$ and $N_i$ be $A$-modules ($i=1,2)$. If $\alpha_1:M_1\to N_1$ and $\alpha_2:M_2\to N_2$ are $A$-linear, then we get a unique $A$-linear map $$\alpha_1\otimes\alpha_2 : M_1\otimes_A M_2\to N_1\otimes N_2$$ which sends $$m_1\otimes m_2 \mapsto \alpha_1(m_1)\otimes\alpha_2(m_2)$$ for all $m_i \in M_i.$ Question: How much can we say about the […]

The definition I have of a tensor product of vector finite dimensional vector spaces $V,W$ over a field $F$ is as follows: Let $v_1, …, v_m$ be a basis for $V$ and let $w_1,…,w_n$ be a basis for $W$. We define $V \otimes W$ to be the set of formal linear combinations of the mn […]

This question already has an answer here: Decomposition of the tensor product $\mathbb{Q}_p \otimes_{\mathbb{Q}} \mathbb{Q}[i]$ into a product of fields 1 answer

Let $k$ be a field, $A$ be a finite dimensional $k$-algebra (say of dimension $n_A$) and let $B\subset A$ be a sub-algebra. What can be said about the $k$-dimension of $A \otimes_B A$ ? The easiest case is when $A$ is $B$-free of rank $r_A$. Then $A \otimes_B A$ has $B$-rank $r_A^2$, and $k$-dimension $n_B^{r_A^2}$ […]

I’m carious to see applications of tensor product. Is there any set of things that if they happen or we encounter them then we use tensor product, tensor algebra, …? I will be happy if it be explained beside an example.

This post is related to the following question: A counterexample to the isomorphism $M^{*} \otimes M \rightarrow Hom_R( M,M)$. I have been trying to isolate what hypothesis can be eliminated in these questions surrounding the tensor products of homomorphism groups and I was wondering if the following questions was formulated in such a way as […]

differential geometer in training here. With regards to my background, I learned differential and Riemannian geometry from O’Neill and Lee’s series. I’m working on my algebra background (which is admittedly a bit weak) and trying to think algebraically about some of the constructions I’m familiar with in differential geometry. If $V$ is an $R$-module over […]

All rings are commutative with unit. Let $\rho:A\rightarrow B$ be a ring homomorphism. Suppose $\mathfrak q$ is a prime ideal of $B$, and let $\mathfrak p=\rho^{-1}(\mathfrak q)$. My question: Is $B_\mathfrak q$ a localization of $B\otimes_A A_\mathfrak p$, or is it equal to $B\otimes_A A_\mathfrak p$? I suspect they are equal, but the book I […]

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