How to get the following $A$-module $Y$ is flat?

Question

Let $A$ be a subalgebra of an algebra $B$ such that $J_A=J_B$(here $J_A$ is the Jacobson radical of $A$), then there is an functor $F=Hom_B(B,-): mod(B) \rightarrow mod(A)$. Now let X be a $B$-module, $Y=FX$, then I have seen that there is an exact sequence $$0 \rightarrow Y \rightarrow B\otimes_AY \rightarrow B/A \otimes_A Y \rightarrow 0$$ of $A$-modules. Obviously we have an exact sequence $$0 \rightarrow A \rightarrow B \rightarrow B/A \rightarrow 0,$$ so to get the first exact sequence, I think we need $Y$ is a flat $A$-module. How to get $Y$ is flat for any $B$-module $X$? Thank you for any help.