Counter-example for a tensor equality.

Question

This is kind of a follow-up to this question, which I fear is wrongly stated (as pointed out by @egreg).

The question is: What is an example of a ring $k$, a ring $R$ which is a $k$-algebra and which is flat as a $k$-module, a $R$-module $B$ and a $k$-module $C$ such that

$$R \otimes_k B \otimes_k C $$ is not isomorphic to $$B \otimes_R R \otimes_k C ?$$

Answer 1

An example for $R \otimes_k B \not\cong B \cong R \otimes_R B$ is enough.

Considering $k= \mathbb Z$ and $R=B=\mathbb Z \times \mathbb Z$ (with pointswise multiplication/addition and obvious scalar multiplication) we get as counterexample:

$$R \otimes_\mathbb{Z} B \cong \mathbb Z^4 \not\cong \mathbb Z^2 =B \quad \text{(as } \mathbb Z \text{-modules)}$$

Since $R$ is a free $\mathbb Z$-module, it is flat.