Searching for some properties of injective modules I have come across this question: Is it true that Tensor product of injective modules is injective?
Unfortunately in that question, there are no proofs or examples. I would like to know:
- How to prove that if $M$ and $N$ are injective $R$-modules, with $R$ PID, $M \otimes_R N$ is injective.
- How to construct a counterexample when $R$ is not a PID.
My attempt:
- Over a PID a module is injective $\iff$ divisible. So I have to show that a tensor product of divisible modules is divisible. Fix $r \in R$ and let $*_r: M \otimes_R N \rightarrow M \otimes_R N$ be the multiplication by $r$. Let $t\in M \otimes_R N \Rightarrow t=\sum_{k=1}^n s_k \cdot m_k \otimes n_k$. Since $M$ is divisible $\exists m_k'\ $ s.t. $\ rm'_k=m_k. \ $ So $t'=\sum_{k=1}^n s_k \cdot m_k' \otimes n_k$ is s.t. $*_r(t')=t$. So $M \otimes_R N$ is divisible and hence injective.
Is the proof correct? I am not sure because I haven't used the injectivity of $N$ in the proof.
- For the counterexample I have no ideas. The only thing I know is that by the previous point $M \otimes N$ will be a divisible module. So the counterexample will be a divisible not injective module. The only example of a divisible not injective module that I know is $\mathbb{Q}(x)/\mathbb{Z}[x]$ as $\mathbb{Z}[x]$ module, but I don't think it can be written as a tensor product of two injective modules.
Thanks a lot.