$R$ be a commutative ring , $A$ be an $R$-algebra , $M$ be an $A$-module , $M_m$ is flat over $R_{m\cap R}$ for every $m\in max(A)$

Question

Let $R$ be a commutative ring , $A$ be an $R$-algebra , $M$ be an $A$-module . If for every maximal ideal $m$ of $A$ , we have $M_m$ is flat over $R_{m\cap R}$ ; then how to show that $M$ is flat over $R$ ?

I know that it is enough to show that $M_p$ is flat over $R_p$ for every maximal ideal $p$ of $R$ ; can we show this from my conditions , or is there any other way to proceed ? Please help . Thanks in advance