Theorems · Theorem · commutative algebra
Submodule.fg_top
∀ {R : Type u_1} {M : Type u_2} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]
(N : Submodule R M), ⊤.FG ↔ N.FGThe top submodule of another submodule N is FG iff N is FG.
See also Module.Finite.fg_top.
- Defined in
- Mathlib.RingTheory.Finiteness.Basic
- Cited by
- 8 results in Mathlib
- Foundations
- Depth 82 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- SemiringAddCommMonoidModule
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
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
- Semiringstatement and proof · cited by 13,802
- AddCommMonoidstatement and proof · cited by 12,281
- Top.topstatement · cited by 9,680
- Submodulestatement and proof · cited by 7,192
- Submodule.subtypeproof · cited by 480
- Subtype.val_injectiveproof · cited by 232
- Submodule.FGstatement and proof · cited by 230
- Submodule.map_topproof · cited by 105
- Submodule.range_subtypeproof · cited by 82
- Submodule.fg_map_iffproof · cited by 1
Cited by8
Results whose statement or proof uses this declaration.
- Module.Finite.iff_fgproof · cited by 21
- isIntegral_transproof · cited by 15
- IsLocalRing.rank_cotangentSpace_eq_spanrank_maximalIdeal_of_fgproof · cited by 1
- TensorProduct.spanFinrank_top_le_of_fgproof · cited by 1
- Algebra.ZariskisMainProperty.quasiFiniteAtproof · cited by 1
- Algebra.Extension.Cotangent.finiteproof · cited by 1
- Algebra.IsUnramifiedAt.exists_hasStandardEtaleSurjectionOnproof · cited by 1
- isIntegral_of_submodule_noetherianproof · cited by 0