Theorems · Theorem · commutative algebra
isNoetherian_def
∀ {R : Type u_1} {M : Type u_2} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M],
IsNoetherian R M ↔ ∀ (s : Submodule R M), s.FGAn R-module is Noetherian iff all its submodules are finitely-generated.
- Defined in
- Mathlib.RingTheory.Noetherian.Defs
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 56 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- SemiringAddCommMonoidModule
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites7
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Modulestatement and proof · cited by 20,661
- Semiringstatement and proof · cited by 13,802
- AddCommMonoidstatement and proof · cited by 12,281
- Submodulestatement · cited by 7,192
- Submodule.FGstatement · cited by 230
- IsNoetherianstatement and proof · cited by 208
- IsNoetherian.noetherianproof · cited by 32
Cited by1
Results whose statement or proof uses this declaration.
- isNoetherianRing_iff_ideal_fgproof · cited by 1