Theorems · Theorem · ring theory
isArtinianRing_iff_isNoetherianRing_krullDimLE_zero
∀ {R : Type u_3} [inst : CommRing R], IsArtinianRing R ↔ IsNoetherianRing R ∧ Ring.KrullDimLE 0 R[Stacks Tag 00KH](https://stacks.math.columbia.edu/tag/00KH)
- Defined in
- Mathlib.RingTheory.HopkinsLevitzki
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 100 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- CommRing
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites5
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- CommRingstatement and proof · cited by 17,173
- IsNoetherianRingstatement and proof · cited by 268
- IsArtinianRingstatement and proof · cited by 98
- Ring.KrullDimLEstatement and proof · cited by 79
- IsNoetherianRing.isArtinianRing_of_krullDimLE_zeroproof · cited by 2
Cited by3
Results whose statement or proof uses this declaration.
- Algebra.QuasiFinite.iff_finite_comap_preimage_singletonproof · cited by 4
- isArtinianRing_iff_krullDimLE_zeroproof · cited by 4
- Module.finite_iff_krullDimLE_zeroproof · cited by 0