Theorems · Theorem · commutative algebra
Ideal.primaryComponent_mem
∀ {A : Type u_1} (M : Type u_2) [inst : CommRing A] (I : Ideal A) [inst_1 : AddCommMonoid M] [inst_2 : Module A M]
(x : M), x ∈ Ideal.primaryComponent M I ↔ ∃ n, x ∈ Submodule.torsionBySet A M ↑(I ^ n)- Cited by
- 0 results in Mathlib
- Foundations
- Depth 78 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- CommRingAddCommMonoidModule
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Cites11
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
- AddCommMonoidstatement and proof · cited by 12,281
- SetLike.coestatement and proof · cited by 8,199
- Submodulestatement · cited by 7,192
- Idealstatement and proof · cited by 4,748
- iSupproof · cited by 2,415
- Submodule.torsionBySetstatement and proof · cited by 34
- Submodule.mem_iSup_of_memproof · cited by 20
- Ideal.primaryComponentstatement · cited by 10
- Submodule.mem_iSup_of_directedproof · cited by 7
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