Theorems · Definition · algebraic geometry
PrimeSpectrum.isIdempotentElemEquivClopens
{R : Type u} → [inst : CommRing R] → { e // IsIdempotentElem e } ≃o TopologicalSpace.Clopens (PrimeSpectrum R)Clopen subsets in the prime spectrum of a commutative ring are in 1-1 correspondence with idempotent elements in the ring.
- Cited by
- 10 results in Mathlib
- Foundations
- Depth 93 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.
Cites8
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
- OrderIsostatement · cited by 874
- PrimeSpectrumstatement · cited by 625
- IsIdempotentElemstatement · cited by 217
- TopologicalSpace.Clopensstatement · cited by 53
- OrderIso.transproof · cited by 31
- OrderIso.isIdempotentElemMulZeroAddOneproof · cited by 0
- PrimeSpectrum.mulZeroAddOneEquivClopensproof · cited by 0
Cited by10
Results whose statement or proof uses this declaration.
- PrimeSpectrum.isIdempotentElemEquivClopens_apply_toOpensstatement · cited by 0
- PrimeSpectrum.isIdempotentElemEquivClopens_mulstatement and proof · cited by 0
- PrimeSpectrum.isIdempotentElemEquivClopens_one_substatement and proof · cited by 0
- PrimeSpectrum.isIdempotentElemEquivClopens_symm_botstatement and proof · cited by 0
- PrimeSpectrum.isIdempotentElemEquivClopens_symm_complstatement and proof · cited by 0
- PrimeSpectrum.isIdempotentElemEquivClopens_symm_infstatement and proof · cited by 0
- PrimeSpectrum.coe_isIdempotentElemEquivClopens_applystatement · cited by 0
- PrimeSpectrum.isIdempotentElemEquivClopens_symm_supstatement and proof · cited by 0
- PrimeSpectrum.isIdempotentElemEquivClopens_symm_topstatement and proof · cited by 0
- PrimeSpectrum.basicOpen_isIdempotentElemEquivClopens_symmstatement and proof · cited by 0