Theorems · Theorem · commutative algebra
Ideal.mem_iInf_smul_pow_eq_bot_iff
∀ {R : Type u_1} {M : Type u_2} [inst : CommRing R] [inst_1 : AddCommGroup M] [inst_2 : Module R M] (I : Ideal R)
[IsNoetherianRing R] [Module.Finite R M] (x : M), x ∈ ⨅ i, I ^ i • ⊤ ↔ ∃ r, ↑r • x = x- Defined in
- Mathlib.RingTheory.Filtration
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 124 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites32
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.topstatement and proof · cited by 9,680
- Submodulestatement and proof · cited by 7,192
- Idealstatement and proof · cited by 4,748
- LE.le.transproof · cited by 3,151
- le_reflproof · cited by 2,061
- iInfstatement and proof · cited by 1,690
- add_commproof · cited by 1,535
- pow_zeroproof · cited by 1,094
- Module.Finitestatement and proof · cited by 1,032
Cited by2
Results whose statement or proof uses this declaration.
- Ideal.iInf_pow_smul_eq_bot_of_isTorsionFreeproof · cited by 3
- Ideal.iInf_pow_smul_eq_bot_of_le_jacobsonproof · cited by 2