Let $A,B$ be commutative rings with 1, $\varphi:B\rightarrow A$ a ring homomorphism and M an $A$-module. If M is flat as an $A$-module, is it also flat as a $B$-module? (The structure of $B$-module is obviously given by $bm=\varphi(b)m$ for all $b\in B,m\in M$).
Suppose $B=\mathbb C[X]$, $A=\mathbb C$ and the map $\phi:B\to A$ is the unique $\mathbb C$-linear ring map such that $\phi(X)=0$.
Now let $M=\mathbb C$ be the free $A$-module of rank $1$, which is plainly flat. Is it $B$-flat?
Another example: $B=\mathbb Z$, $A=\mathbb Z/2\mathbb Z$ and let $M$ be again a free $A$-module of rank $1$.