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

https://stacks.math.columbia.edu/tag/00H9

for a proof. Does the argument extend to the noncommutative setting?