Let $R$ be a possibly noncommutative left Noetherian ring and $M$ an $R$-module. I am looking for a reference or a proof for the following fact: $M$ is finitely generated and projective if and only if it is finitely presented and flat. (I am not interested in references that treat only the commutative case.)



Finitely generated projective = finitely presented flat over a noncommutative Noetherian ring
This holds over any ring, noetherian or not. See Bourbaki *Algebra* X, §1, no. 5.


