Theorems · Theorem · commutative algebra

RingTheory.Sequence.isRegular_iff_isWeaklyRegular_of_subset_jacobson_annihilator

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

Defined in: Mathlib.RingTheory.Regular.RegularSequence
Cited by: 1 results in Mathlib
Foundations: Depth 81 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: CommRing AddCommGroup Module Nontrivial Module.Finite

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Cites13

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
Idealstatement · cited by 4,748
Nontrivialstatement and proof · cited by 2,416
Module.Finitestatement and proof · cited by 1,032
Ideal.jacobsonstatement and proof · cited by 88
Ideal.span_leproof · cited by 70
Module.annihilatorstatement and proof · cited by 61
RingTheory.Sequence.IsWeaklyRegularstatement · cited by 40
RingTheory.Sequence.IsRegularstatement · cited by 29
RingTheory.Sequence.isRegular_iffproof · cited by 6

Cited by1

Results whose statement or proof uses this declaration.

IsLocalRing.isRegular_iff_isWeaklyRegular_of_subset_maximalIdealproof · cited by 1