Articles of tensor products

$I\otimes I$ is torsion free for a principal ideal $I$ in domain $R$

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 […]

$\mathbb{C}\otimes_\mathbb{C} \mathbb{C} \cong \mathbb{R}\otimes _\mathbb{R} \mathbb{C}$

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

Description of the kernel of the tensor product of two linear maps

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 […]

Understanding definition of tensor product

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 […]

Computing the quotient $\mathbb{Q}_p/(x^2 + 1)$

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

dimension of tensor products over a submodule

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 $n_B^{r_A^2}$ […]

When do we use Tensor?

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.

Isomorphism involving tensor products of homomorphism groups

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 […]

What sort of algebraic structure describes the “tensor algebra” of tensors of mixed variance in differential geometry?

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 […]

Quick question on localization of tensor products

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 […]