Theorems · Theorem · commutative algebra
Ideal.Filtration.submodule_fg_iff_stable
∀ {R : Type u_1} {M : Type u_2} [inst : CommRing R] [inst_1 : AddCommGroup M] [inst_2 : Module R M] {I : Ideal R}
(F : I.Filtration M), (∀ (i : ℕ), (F.N i).FG) → (F.submodule.FG ↔ F.Stable)If the components of a filtration are finitely generated, then the filtration is stable iff its associated submodule of is finitely generated.
- Defined in
- Mathlib.RingTheory.Filtration
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 121 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- CommRingAddCommGroupModule
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites46
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coeproof · cited by 62,936
- Setproof · cited by 53,352
- Modulestatement and proof · cited by 20,661
- CommRingstatement and proof · cited by 17,173
- Finsetproof · cited by 13,712
- AddCommGroupstatement and proof · cited by 12,871
- SetLike.coeproof · cited by 8,199
- Submoduleproof · cited by 7,192
- Polynomialstatement and proof · cited by 5,681
- Set.imageproof · cited by 5,609
- Idealstatement and proof · cited by 4,748
- LE.le.transproof · cited by 3,151
Cited by1
Results whose statement or proof uses this declaration.
- Ideal.Filtration.Stable.of_leproof · cited by 2