# Finitely generated projective = finitely presented flat over a noncommutative Noetherian ring

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.)

They show that the Cauchy completion of a small $\mathcal{V}$-category $\mathcal{C}$ is equivalent to the category of $\mathcal{V}$-finitely presentable $\mathcal{V}$-flat presheaves on $\mathcal{C}$.
To get the result you ask about, take $\mathcal{V} = \textbf{Ab}$ (the category of abelian groups) and treat a ring $R$ as a $\mathcal{V}$-enriched category $\mathcal{R}$ with one object. Then use the fact that the Cauchy completion of $\mathcal{R}$ is the category of finitely generated projective $R$ modules.