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?
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. 


Ref, H. Matsumura, Commutative Ring Theory P1620 probably... The counter example is just @Sasha mentioned. 

