Theorems · Theorem · commutative algebra
Module.Finite.of_fg
∀ {R : Type u_1} {M : Type u_4} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]
{N : Submodule R M}, N.FG → Module.Finite R ↥NA finitely-generated submodule is finite as a module.
- Defined in
- Mathlib.RingTheory.Finiteness.Basic
- Cited by
- 21 results in Mathlib
- Foundations
- Depth 84 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.
Cites7
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
- Submodulestatement and proof · cited by 7,192
- Module.Finitestatement · cited by 1,032
- Submodule.FGstatement · cited by 230
- Module.Finite.iff_fgproof · cited by 21
Cited by21
Results whose statement or proof uses this declaration.
- isIntegral_transproof · cited by 15
- IsIntegral.of_mem_of_fgproof · cited by 13
- Module.Finite.span_of_finiteproof · cited by 6
- ZLattice.module_finiteproof · cited by 3
- LinearMap.HasFiniteRange.hasNoetherianRangeproof · cited by 3
- IsIntegral.inv_mem_adjoinproof · cited by 2
- Submodule.finite_quotient_smulproof · cited by 2
- Module.Flat.exists_factorization_of_finitePresentationproof · cited by 2
- Subalgebra.LinearDisjoint.of_linearDisjoint_finite_leftproof · cited by 2
- Module.exists_basis_of_basis_baseChangeproof · cited by 1
- Algebra.FormallySmooth.iff_injective_lTensor_residueFieldproof · cited by 1
- IsIntegral.isUnitproof · cited by 1