Theorems · Theorem · commutative algebra
isNoetherian_submodule
∀ {R : Type u_1} {M : Type u_2} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]
{N : Submodule R M}, IsNoetherian R ↥N ↔ ∀ s ≤ N, s.FG- Defined in
- Mathlib.RingTheory.Noetherian.Defs
- Cited by
- 8 results in Mathlib
- Foundations
- Depth 81 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.
Cites16
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 and proof · cited by 7,192
- LinearMap.rangeproof · cited by 893
- Submodule.mapproof · cited by 614
- Submodule.subtypeproof · cited by 480
- Submodule.comapproof · cited by 347
- Subtype.val_injectiveproof · cited by 232
- Submodule.FGstatement and proof · cited by 230
- IsNoetherianstatement and proof · cited by 208
- Submodule.range_subtypeproof · cited by 82
Cited by8
Results whose statement or proof uses this declaration.
- FractionalIdeal.isNoetherian_iffproof · cited by 2
- Ideal.Filtration.Stable.of_leproof · cited by 2
- isNoetherian_of_leproof · cited by 2
- Submodule.FG.of_le_of_isNoetherianproof · cited by 1
- FractionalIdeal.isNoetherian_zeroproof · cited by 1
- isNoetherian_submodule_leftproof · cited by 1
- FractionalIdeal.coe_ideal_mul_invproof · cited by 1
- isNoetherian_submodule_rightproof · cited by 0