Let $R$ be a not necessarily commutative, unital, ring, and for simplicity let module always mean right module. We say that a unital ring extension $R \hookrightarrow S$ is flat, or faithfully flat, if $S$ is flat, or respectively faithfully flat, as an $R$-module.
Is the composition of two flat, faithfully flat, ring extensions again flat, respectively faithfully flat?
Edit: It seems that in the commutative case this is true. See
for a proof. Does the argument extend to the noncommutative setting?