Tensor product over different rings are not equal I'm looking for an example of modules $A,B$ such that $A\otimes_{R}B\neq A\otimes_{\mathbb{Z}}B$.
I've thought about some tensor products I know but I haven't been able to produce an example. What I've considered are as follows: $\mathbb{Z}_{n}\otimes\mathbb{Z}_{m}\cong\mathbb{Z}_{\gcd(n,m)}$, $\mathbb{Q}\otimes\mathbb{Q}\cong\mathbb{Q}$, $A\otimes\mathbb{Q}=0$ for any torsion abelian group $A$, $A\otimes_{R}R\cong A$ if $R$ is a ring with 1.

Hint: look at the last example you mention. It may help to pick a special $A$....
Incidentally, this turns out to be one of those things that, in the grand scheme of things, getting equality of the two tensor products is a very rare thing -- it's just that you get equality for a number of "nice" constructions, and so if you only consider the nice ones you won't find an example.
