An example of two tensor products on different rings being not isomorphic

Question

I want to find two rings $R_1,R_2$ and two abelian groups $A,B$ such that they are both $R_1$ and $R_2$ modules and $A\otimes_{R_1} B$ is nonisomorphic to $A\otimes_{R_2} B$. I am trying to do it by using either of the two facts 1) For any unitary $R$-module $A$ we have $A\otimes_R R= A$, and 2) if $A$ is a torsion abelian group then $A\otimes \mathbb Q=0$. However, I could not make it. How should I find such an example using either of those two facts?

Answer 1

For example you can consider $R_1=\mathbb{Z}$, $R_2=\mathbb{Q}$, $B=\mathbb{Q}$. In this case you have that

1. $A\otimes_\mathbb{Z} \mathbb{Q}$ is the free group with base A ;

2. $A\otimes_\mathbb{Q}\mathbb{Q}\cong A$