theorem submodule_fg_of_noetherian_fg (R : Type u_1) [CommRing R] [IsNoetherianRing R] (M : Type u_2) [AddCommGroup M] [Module R M] [Module.Finite R M] (N : Submodule R M) : N.FG

Over a Noetherian ring, any submodule of a finitely generated module is itself finitely generated.

theorem subsheaf_coherent_of_noetherian (R : Type u_1) [CommRing R] [IsNoetherianRing R] (M : Type u_2) [AddCommGroup M] [Module R M] (hM : Module.FinitePresentation R M) (N : Submodule R M) : Module.FinitePresentation R ↥N

Lec 12 (coherent subsheaves): over a Noetherian ring, any submodule of a finitely presented module is again finitely presented, the algebraic analogue of "a subsheaf of a coherent sheaf is coherent".