Theorems · Theorem · commutative algebra
IsNoetherianRing.of_prime
∀ {R : Type u_1} [inst : CommRing R], (∀ (I : Ideal R), I.IsPrime → I.FG) → IsNoetherianRing RIf all prime ideals in a commutative ring are finitely generated, so are all other ideals.
- Defined in
- Mathlib.RingTheory.Noetherian.OfPrime
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 86 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.
Cites21
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setproof · cited by 53,352
- CommRingstatement and proof · cited by 17,173
- Finsetproof · cited by 13,712
- SetLike.coeproof · cited by 8,199
- Set.ofPredproof · cited by 6,101
- Idealstatement and proof · cited by 4,748
- le_antisymmproof · cited by 2,068
- SupSet.sSupproof · cited by 954
- Ideal.spanproof · cited by 948
- Ideal.IsPrimestatement and proof · cited by 827
- IsNoetherianRingstatement · cited by 268
- IsChainproof · cited by 158
Cited by1
Results whose statement or proof uses this declaration.
- IsNoetherianRing.of_prime_ne_botproof · cited by 0