Theorems · Theorem · commutative algebra

IsLocalRing.isRegular_iff_isWeaklyRegular_of_subset_maximalIdeal

∀ {R : Type u_1} {M : Type u_3} [inst : CommRing R] [inst_1 : AddCommGroup M] [inst_2 : Module R M]
  [inst_3 : IsLocalRing R] [Nontrivial M] [Module.Finite R M] {rs : List R},
  (∀ r ∈ rs, r ∈ IsLocalRing.maximalIdeal R) →
    (RingTheory.Sequence.IsRegular M rs ↔ RingTheory.Sequence.IsWeaklyRegular M rs)

Defined in: Mathlib.RingTheory.Regular.RegularSequence
Cited by: 1 results in Mathlib
Foundations: Depth 82 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: CommRing AddCommGroup Module IsLocalRing Nontrivial Module.Finite

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Cites17

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

Modulestatement and proof · cited by 20,661
CommRingstatement and proof · cited by 17,173
AddCommGroupstatement and proof · cited by 12,871
Top.topproof · cited by 9,680
Idealstatement · cited by 4,748
Nontrivialstatement and proof · cited by 2,416
Module.Finitestatement and proof · cited by 1,032
IsLocalRingstatement and proof · cited by 339
IsLocalRing.maximalIdealstatement and proof · cited by 297
Module.annihilatorproof · cited by 61
RingTheory.Sequence.IsWeaklyRegularstatement · cited by 40
RingTheory.Sequence.IsRegularstatement · cited by 29

Cited by1

Results whose statement or proof uses this declaration.

RingTheory.Sequence.IsWeaklyRegular.isRegular_of_isLocalizedModule_of_memproof · cited by 1