Chapter 9, Tensor Products Video Solutions, A Course in Ring Theory

Problem 1

Suppose $I \triangleleft R$ and that $A_{R}$ and $_{R} B$ are $R$-modules with $A I=0=I B$. Prove that $A \otimes_{(R / I)} B$ is naturally isomorphic to $A \otimes_{R} B$.

Doruk Isik

Numerade Educator

02:24

Problem 2

Let $I$ be a left ideal of $R$ and $J$ a right ideal. If $_{R} I$ is flat, prove that $J \otimes I \cong J I$. If $_{R}(R / I)$ is flat, show that $J \cap I=J I$.

M Hassan Anwar

Numerade Educator

02:48

If all $R$-modules are flat, prove that $R$ is von Neumann regular. Conversely if $R$ is von Neumann regular, show at least that all submodules of free $R$-modules are flat. For the first part, choose $r \in R$ and apply the preceding exercise with $I=R r$ and $J=r R$. For the converse, use the fact that $R$ is semihereditary.

Mohan Jain

Numerade Educator

02:45

Problem 4

Let $R$ be the polynomial ring $K[x, y]$ and let $I=R x+R y$. Show that ${ }_{R} I$ is not flat. For this, first observe that if $I$ is flat, then $I^{2} \cong I \otimes I$ as $(R, R)$-bimodules. Next note that $I^{2} / I^{3}$ is the largest homomorphic image of $I^{2}$ that is annihilated on both sides by $I$ and that $\operatorname{dim}_{K} I^{2} / I^{3}=3$. On the other hand, show that $I \otimes I$ maps onto $\left(I / I^{2}\right) \otimes_{R}\left(I / I^{2}\right)$ and that this has dimension $4 .$

Dharmendra Jain

Numerade Educator

01:23

Problem 5

Show that the direct sum of a flat module and a faithfully flat module is faithfully flat. Conclude, therefore, that any nonzero free module is faithfully flat. Find an example of a flat $\mathbb{Z}$-module that is not faithfully flat.

Joseph Liao

Numerade Educator

Problem 6

Show that ${ }_{R} B$ is faithfully flat if and only if, for all maps $\alpha: A \rightarrow A^{\prime}$ either $\alpha$ or $\alpha \otimes 1_{B}$ being a monomorphism implies that the other is also. This is a key alternate characterization of faithfully flat modules.

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04:14

Problem 7

A family $\mathcal{M}$ of submodules of $B$ is said to be directed if its members generate $B$ and if every two members are contained in a third. If $\mathcal{M}$ is such a directed family, show that $B=\bigcup_{M \in \mathcal{M}} M$ and that $\mathcal{M}$ is a local system.

Sirat Shah

Numerade Educator

03:38

Problem 8

Let $A_{R}^{\prime} \subseteq A_{R}$ and $_{R} B$ be given. Show that there exists a submodule $L$ of $A$ containing $A^{\prime}$ that is maximal with respect to $A^{\prime} \otimes B \rightarrow$ $L \otimes B$ being a monomorphism. Similarly, show that there exists a submodule $M$ of $B$ maximal with respect to $A^{\prime} \otimes M \rightarrow A \otimes M$ being a monomorphism.

Prathan Jarupoonphol

Numerade Educator

08:50

Problem 9

Let $K$ be a field and let $R$ and $S$ be $K$-algebras. We consider the tensor product $R \otimes_{K} S$. To start with, define a multiplication in $\mathcal{S}(R, S)$ distributively by $\left(r_{1}, s_{1}\right) \cdot\left(r_{2}, s_{2}\right)=\left(r_{1} r_{2}, s_{1} s_{2}\right) .$ Then show that, in this way, $\mathcal{S}(R, S)$ is an associative ring and that $\mathcal{S}_{0}(R, S)$ is a two-sided ideal. Conclude that $R \otimes_{K} S$ is an associative $K$-algebra with multiplication given by $\left(r_{1} \otimes s_{1}\right) \cdot\left(r_{2} \otimes s_{2}\right)=\left(r_{1} r_{2}\right) \otimes\left(s_{1} s_{2}\right)$.