Theorems · Theorem · commutative algebra

Module.finite_def

∀ {R : Type u_6} {M : Type u_7} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M],
  Module.Finite R M ↔ ⊤.FG

See also Module.Finite.iff_fg for a version when M is itself a submodule.

Defined in: Mathlib.RingTheory.Finiteness.Defs
Cited by: 16 results in Mathlib
Foundations: Depth 56 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: Semiring AddCommMonoid Module

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Cites8

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 · cited by 7,192
Module.Finitestatement and proof · cited by 1,032
Submodule.FGstatement · cited by 230
Module.Finite.fg_topproof · cited by 28

Cited by16

Results whose statement or proof uses this declaration.

Module.Finite.of_surjectiveproof · cited by 29
Module.rank_lt_aleph0proof · cited by 23
Module.Finite.iff_fgproof · cited by 21
Module.Finite.of_restrictScalars_finiteproof · cited by 12
Module.Finite.smallproof · cited by 3
IsCyclotomicExtension.finite_of_singletonproof · cited by 2
IsLocalRing.adjoin_residue_eq_top_iff_adjoin_eq_topproof · cited by 2
LinearMap.finsuppLinearMap_bijective_of_moduleFiniteproof · cited by 1
IsCyclotomicExtension.finiteproof · cited by 1
Polynomial.not_finiteproof · cited by 1
IsLocalRing.quotient_span_eq_top_iff_span_eq_topproof · cited by 1
LinearMap.surjective_of_surjective_comp_mkQproof · cited by 1