Theorems · Theorem · commutative algebra
Submodule.fg_iff_exists_fin_generating_family
∀ {R : Type u_1} {M : Type u_2} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]
{N : Submodule R M}, N.FG ↔ ∃ n s, Submodule.span R (Set.range s) = N- Defined in
- Mathlib.RingTheory.Finiteness.Defs
- Cited by
- 6 results in Mathlib
- Foundations
- Depth 80 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.
Cites14
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
- Semiringstatement and proof · cited by 13,802
- AddCommMonoidstatement and proof · cited by 12,281
- Submodulestatement and proof · cited by 7,192
- Set.rangestatement and proof · cited by 4,705
- Set.Finiteproof · cited by 1,814
- Submodule.spanstatement and proof · cited by 1,504
- Function.Embeddingproof · cited by 988
- Submodule.FGstatement · cited by 230
- Set.finite_rangeproof · cited by 58
Cited by6
Results whose statement or proof uses this declaration.
- Module.Finite.exists_finproof · cited by 5
- PowerSeries.eq_of_le_of_X_notMem_of_fg_of_isPrimeproof · cited by 1
- Submodule.fg_iff_exists_finite_generating_familyproof · cited by 1
- Submodule.FG.smallproof · cited by 1
- Ideal.isOka_fgproof · cited by 1
- Algebra.Presentation.exists_presentation_finproof · cited by 0