I am in the condition where I have a noetherian ring $R$ of finite global dimension. Consider the category of finitely generated (right) modules over $R$. Then I want to show that every module admits a finite projective resolution of finitely generated projective modules.
The definition of being having finite global dimension only means that all modules have a finite projective resolution, I do not know how to prove that the resolution is finitely generated.
Any hint?
Thanks!