Theorems · Theorem · commutative algebra
Module.Finite.span_of_finite
∀ (R : Type u_1) {M : Type u_4} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {A : Set M},
A.Finite → Module.Finite R ↥(Submodule.span R A)The submodule generated by a finite set is R-finite.
- Defined in
- Mathlib.RingTheory.Finiteness.Basic
- Cited by
- 6 results in Mathlib
- Foundations
- Depth 85 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.
- Setstatement and proof · 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 · cited by 7,192
- Set.Finitestatement and proof · cited by 1,814
- Submodule.spanstatement · cited by 1,504
- Module.Finitestatement · cited by 1,032
- Set.Finite.toFinsetproof · cited by 351
- Set.Finite.coe_toFinsetproof · cited by 124
- Module.Finite.of_fgproof · cited by 21
Cited by6
Results whose statement or proof uses this declaration.
- FiniteDimensional.of_totallyBounded_nhds_zeroproof · cited by 3
- Matrix.rank_submatrix_leproof · cited by 1
- Projectivization.isCollinear_pairproof · cited by 1
- Module.End.eq_zero_of_isNilpotent_of_isFinitelySemisimpleproof · cited by 1
- FiniteDimensional.span_of_finiteproof · cited by 1
- Projectivization.isCollinear_singleton'proof · cited by 0