Theorems · Theorem · commutative algebra

IsLocalRing.quotient_span_eq_top_iff_span_eq_top

∀ {R : Type u_1} {S : Type u_2} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S]
  [inst_3 : IsLocalRing R] [Module.Finite R S] (s : Set S),
  Submodule.span (R ⧸ IsLocalRing.maximalIdeal R)
        (⇑(Ideal.Quotient.mk (Ideal.map (algebraMap R S) (IsLocalRing.maximalIdeal R))) '' s) =
      ⊤ ↔
    Submodule.span R s = ⊤

Defined in: Mathlib.RingTheory.LocalRing.Quotient
Cited by: 1 results in Mathlib
Foundations: Depth 92 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: CommRing CommRing Algebra IsLocalRing Module.Finite

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Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

DFunLike.coestatement and proof · cited by 62,936
Setstatement and proof · cited by 53,352
CommRingstatement and proof · cited by 17,173
Algebrastatement and proof · cited by 11,388
RingHomstatement and proof · cited by 10,189
Top.topstatement and proof · cited by 9,680
Submodulestatement and proof · cited by 7,192
Set.imagestatement and proof · cited by 5,609
Idealstatement · cited by 4,748
Bot.botproof · cited by 4,720
Algebra.algebraMapstatement and proof · cited by 4,706
HasQuotient.Quotientstatement and proof · cited by 2,301

Cited by1

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IsLocalRing.finrank_quotient_mapproof · cited by 1