I've been doing some tensoring, but am having a hard time understanding the following isomorphism.
Suppose $A$ is a commutative $R$-algebra, and for any $R$-module $M$, denote by $M_A=A\otimes_R M$. Then for any $R$-module and $A$-module $N$, $M_A\otimes_A N\cong M\otimes_R N$.
I tried to explain this to myself by defining some $\varphi_a\colon M\times N\to A\otimes_R M\otimes_A N$ defined by $m\otimes n\mapsto a\otimes m\otimes n$ to get some module homomorphism $M\otimes_R N\to A\otimes_R M\otimes_A N$. I was thinking then I could use some sort of projection $g\colon A\otimes_R M\otimes_A N\to M\otimes_R N$ given by $a\otimes m\otimes n\mapsto m\otimes n$ to find an inverse. However, I'm feeling shaky since the maps seem to depend on a certain $a\in A$, so maybe what I'm saying is nonsense.
What is the proper explanation for this isomorphism?