Theorems · Theorem · ring theory
IsLocalRing.isOpen_iff_finite_quotient
∀ {R : Type u_1} [inst : CommRing R] [inst_1 : TopologicalSpace R] [IsTopologicalRing R] [CompactSpace R] [T2Space R]
[IsLocalRing R] [IsNoetherianRing R] {I : Ideal R}, IsOpen ↑I ↔ Finite (R ⧸ I)- Defined in
- Mathlib.Topology.Algebra.Ring.Compact
- Cited by
- 0 results in Mathlib
- Foundations
- Depth 104 from the axioms · uses propext, Classical.choice, Quot.sound
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Cites18
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- TopologicalSpacestatement and proof · cited by 24,529
- CommRingstatement and proof · cited by 17,173
- SetLike.coestatement · cited by 8,199
- Idealstatement and proof · cited by 4,748
- Finitestatement and proof · cited by 3,029
- IsOpenstatement · cited by 2,400
- HasQuotient.Quotientstatement and proof · cited by 2,301
- T2Spacestatement and proof · cited by 1,351
- CompactSpacestatement and proof · cited by 593
- IsTopologicalRingstatement and proof · cited by 402
- IsLocalRingstatement and proof · cited by 339
- IsLocalRing.maximalIdealproof · cited by 297
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