Theorems · Theorem · commutative algebra
isFiniteLength_iff_isNoetherian_isArtinian
∀ {R : Type u_1} [inst : Ring R] {M : Type u_2} [inst_1 : AddCommGroup M] [inst_2 : Module R M],
IsFiniteLength R M ↔ IsNoetherian R M ∧ IsArtinian R M- Defined in
- Mathlib.RingTheory.FiniteLength
- Cited by
- 14 results in Mathlib
- Foundations
- Depth 93 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- RingAddCommGroupModule
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites14
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
- AddCommGroupstatement and proof · cited by 12,871
- Ringstatement and proof · cited by 7,463
- Submoduleproof · cited by 7,192
- HasQuotient.Quotientproof · cited by 2,301
- IsNoetherianstatement and proof · cited by 208
- IsSimpleModuleproof · cited by 114
- IsArtinianstatement and proof · cited by 69
- IsFiniteLengthstatement and proof · cited by 20
- isNoetherian_iff'proof · cited by 6
- isNoetherian_iff_submodule_quotientproof · cited by 3
- isFiniteLength_of_exists_compositionSeriesproof · cited by 3
Cited by14
Results whose statement or proof uses this declaration.
- Module.length_ne_top_iffproof · cited by 8
- Module.length_compositionSeriesproof · cited by 4
- isFiniteLength_iff_exists_compositionSeriesproof · cited by 4
- Ring.ord_eq_addValproof · cited by 3
- Module.length_of_freeproof · cited by 3
- IsArtinianRing.tfaeproof · cited by 2
- IsSemisimpleModule.finite_tfaeproof · cited by 2
- isArtinianRing_iff_isFiniteLengthproof · cited by 1
- isFiniteLength_quotient_span_singletonproof · cited by 1
- IsLocalRing.length_baseChangeproof · cited by 1
- IsLocalRing.length_restrictScalarsproof · cited by 1
- IsFiniteLength.of_injectiveproof · cited by 1