Let $R\_1$ and $R\_2$ be two subrings of the ring $R$ which commute in $R$ so that we have a ring homomorphism $R\_1\otimes\_\mathbb{Z} R\_2\rightarrow R$. Assume that $R$ is flat over $R\_1$ and $R\_2$. Is then $R$ also flat over $R\_1\otimes\_\mathbb{Z} R\_2$? Is there an easy counterexample?



Flatness and tensor product of rings
Take $R\_1 = R\_2 = R = {\mathbb Z}[x]$. Then $R\_1\otimes\_{\mathbb Z} R\_2 = {\mathbb Z}[x\_1,x\_2]$ and $R = {\mathbb Z}[x]$ is not flat over it.


