Theorems · Theorem · commutative algebra
Ideal.krullDimLE_zero_quotient_iff_forall_minimalPrimes_isMaximal
∀ {R : Type u_2} [inst : CommRing R] {I : Ideal R}, Ring.KrullDimLE 0 (R ⧸ I) ↔ ∀ J ∈ I.minimalPrimes, J.IsMaximalA quotient R ⧸ I has krull dimension at most zero if and only if all minimal primes over I
are maximal.
- Defined in
- Mathlib.RingTheory.KrullDimension.Zero
- Cited by
- 0 results in Mathlib
- Foundations
- Depth 88 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- CommRing
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Cites20
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement and proof · cited by 53,352
- CommRingstatement and proof · cited by 17,173
- Top.topproof · cited by 9,680
- Idealstatement and proof · cited by 4,748
- HasQuotient.Quotientstatement and proof · cited by 2,301
- Ideal.mapproof · cited by 692
- Ideal.Quotient.mkproof · cited by 610
- Ideal.IsMaximalstatement and proof · cited by 452
- Ideal.comapproof · cited by 443
- Ideal.Quotient.mk_surjectiveproof · cited by 134
- Ideal.IsPrime.ne_topproof · cited by 82
- Ring.KrullDimLEstatement · cited by 79
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