Maximal spectrum of a commutative ring

The maximal spectrum of a commutative ring is the type of all maximal ideals. It is naturally a subset of the prime spectrum endowed with the subspace topology.

Main definitions

• MaximalSpectrum R: The maximal spectrum of a commutative ring R, i.e., the set of all maximal ideals of R.

Implementation notes

The Zariski topology on the maximal spectrum is defined as the subspace topology induced by the natural inclusion into the prime spectrum to avoid API duplication for zero loci.

theorem MaximalSpectrum.toPrimeSpectrum_range {R : Type u} [CommRing R] : Set.range MaximalSpectrum.toPrimeSpectrum = {x : PrimeSpectrum R | IsClosed {x}}

instance MaximalSpectrum.zariskiTopology {R : Type u} [CommRing R] : TopologicalSpace (MaximalSpectrum R)

The Zariski topology on the maximal spectrum of a commutative ring is defined as the subspace topology induced by the natural inclusion into the prime spectrum.

Equations

• MaximalSpectrum.zariskiTopology = TopologicalSpace.induced MaximalSpectrum.toPrimeSpectrum PrimeSpectrum.zariskiTopology

instance MaximalSpectrum.instT1Space {R : Type u} [CommRing R] : T1Space (MaximalSpectrum R)

Equations

• ⋯ = ⋯

theorem MaximalSpectrum.toPrimeSpectrum_continuous {R : Type u} [CommRing R] : Continuous MaximalSpectrum.toPrimeSpectrum