Theorems · Theorem · commutative algebra
IsNoetherianRing.of_prime_ne_bot
∀ {R : Type u_1} [inst : CommRing R], (∀ (I : Ideal R), I.IsPrime → I ≠ ⊥ → I.FG) → IsNoetherianRing RIf all non-zero prime ideals in a commutative ring are finitely generated, so are all other ideals.
- Defined in
- Mathlib.RingTheory.Noetherian.OfPrime
- Cited by
- 0 results in Mathlib
- Foundations
- Depth 87 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- CommRing
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Cites9
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
- Idealstatement and proof · cited by 4,748
- Bot.botstatement and proof · cited by 4,720
- eq_or_neproof · cited by 1,117
- Ideal.IsPrimestatement and proof · cited by 827
- IsNoetherianRingstatement · cited by 268
- Ideal.FGstatement and proof · cited by 99
- Submodule.fg_botproof · cited by 13
- IsNoetherianRing.of_primeproof · cited by 1
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