Noetherian rings and modules

The following are equivalent for a module M over a ring R:

1. Every increasing chain of submodules M₁ ⊆ M₂ ⊆ M₃ ⊆ ⋯ eventually stabilises.
2. Every submodule is finitely generated.

A module satisfying these equivalent conditions is said to be a Noetherian R-module. A ring is a Noetherian ring if it is Noetherian as a module over itself.

(Note that we do not assume yet that our rings are commutative, so perhaps this should be called "left Noetherian". To avoid cumbersome names once we specialize to the commutative case, we don't make this explicit in the declaration names.)

Main definitions

Let R be a ring and let M and P be R-modules. Let N be an R-submodule of M.

• IsNoetherian R M is the proposition that M is a Noetherian R-module. It is a class, implemented as the predicate that all R-submodules of M are finitely generated.

Main statements

• isNoetherian_iff is the theorem that an R-module M is Noetherian iff > is well-founded on Submodule R M.

Note that the Hilbert basis theorem, that if a commutative ring R is Noetherian then so is R[X], is proved in RingTheory.Polynomial.

References

• M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra
• P. Samuel, Algebraic Theory of Numbers

Tags

Noetherian, noetherian, Noetherian ring, Noetherian module, noetherian ring, noetherian module

theorem isNoetherian_of_surjective {R : Type u_1} (M : Type u_2) {P : Type u_3} [Semiring R] [AddCommMonoid M] [AddCommMonoid P] [Module R M] [Module R P] (f : M →ₗ[R] P) (hf : LinearMap.range f = ⊤) [IsNoetherian R M] : IsNoetherian R P

instance isNoetherian_range {R : Type u_1} {M : Type u_2} {P : Type u_3} [Semiring R] [AddCommMonoid M] [AddCommMonoid P] [Module R M] [Module R P] (f : M →ₗ[R] P) [IsNoetherian R M] : IsNoetherian R ↥(LinearMap.range f)

instance isNoetherian_quotient {R : Type u_1} [Semiring R] {A : Type u_4} {M : Type u_5} [Ring A] [AddCommGroup M] [SMul R A] [Module R M] [Module A M] [IsScalarTower R A M] (N : Submodule A M) [IsNoetherian R M] : IsNoetherian R (M ⧸ N)

theorem isNoetherian_of_linearEquiv {R : Type u_1} {M : Type u_2} {P : Type u_3} [Semiring R] [AddCommMonoid M] [AddCommMonoid P] [Module R M] [Module R P] (f : M ≃ₗ[R] P) [IsNoetherian R M] : IsNoetherian R P

theorem LinearEquiv.isNoetherian_iff {R : Type u_1} {M : Type u_2} {P : Type u_3} [Semiring R] [AddCommMonoid M] [AddCommMonoid P] [Module R M] [Module R P] (f : M ≃ₗ[R] P) : IsNoetherian R M ↔ IsNoetherian R P

theorem isNoetherian_top_iff {R : Type u_1} {M : Type u_2} [Semiring R] [AddCommMonoid M] [Module R M] : IsNoetherian R ↥⊤ ↔ IsNoetherian R M

theorem isNoetherian_of_injective {R : Type u_1} {M : Type u_2} {P : Type u_3} [Semiring R] [AddCommMonoid M] [AddCommMonoid P] [Module R M] [Module R P] [IsNoetherian R P] (f : M →ₗ[R] P) (hf : Function.Injective ⇑f) : IsNoetherian R M

theorem fg_of_injective {R : Type u_1} {M : Type u_2} {P : Type u_3} [Semiring R] [AddCommMonoid M] [AddCommMonoid P] [Module R M] [Module R P] [IsNoetherian R P] {N : Submodule R M} (f : M →ₗ[R] P) (hf : Function.Injective ⇑f) : N.FG

@[instance 80]

instance isNoetherian_of_finite (R : Type u_1) (M : Type u_2) [Semiring R] [AddCommMonoid M] [Module R M] [Finite M] : IsNoetherian R M

@[instance 100]

instance Module.IsNoetherian.finite (R : Type u_1) (M : Type u_2) [Semiring R] [AddCommMonoid M] [Module R M] [IsNoetherian R M] : Module.Finite R M

instance Module.instFiniteSubtypeMemIdealOfIsNoetherian {R₁ : Type u_4} {S : Type u_5} [CommSemiring R₁] [Semiring S] [Algebra R₁ S] [IsNoetherian R₁ S] (I : Ideal S) : Module.Finite R₁ ↥I

theorem Module.Finite.of_injective {R : Type u_1} {M : Type u_2} {N : Type u_3} [Semiring R] [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N] [IsNoetherian R N] (f : M →ₗ[R] N) (hf : Function.Injective ⇑f) : Module.Finite R M

theorem isNoetherian_of_ker_bot {R : Type u_1} {M : Type u_2} {P : Type u_4} [Ring R] [AddCommGroup M] [AddCommGroup P] [Module R M] [Module R P] [IsNoetherian R P] (f : M →ₗ[R] P) (hf : LinearMap.ker f = ⊥) : IsNoetherian R M

theorem fg_of_ker_bot {R : Type u_1} {M : Type u_2} {P : Type u_4} [Ring R] [AddCommGroup M] [AddCommGroup P] [Module R M] [Module R P] [IsNoetherian R P] {N : Submodule R M} (f : M →ₗ[R] P) (hf : LinearMap.ker f = ⊥) : N.FG

instance isNoetherian_prod {R : Type u_1} {M : Type u_2} {P : Type u_4} [Ring R] [AddCommGroup M] [AddCommGroup P] [Module R M] [Module R P] [IsNoetherian R M] [IsNoetherian R P] : IsNoetherian R (M × P)

instance isNoetherian_sup {R : Type u_1} {P : Type u_4} [Ring R] [AddCommGroup P] [Module R P] (M₁ M₂ : Submodule R P) [IsNoetherian R ↥M₁] [IsNoetherian R ↥M₂] : IsNoetherian R ↥(M₁ ⊔ M₂)

instance isNoetherian_pi {R : Type u_1} [Ring R] {ι : Type u_5} [Finite ι] {M : ι → Type u_6} [(i : ι) → AddCommGroup (M i)] [(i : ι) → Module R (M i)] [∀ (i : ι), IsNoetherian R (M i)] : IsNoetherian R ((i : ι) → M i)

instance isNoetherian_pi' {R : Type u_1} {M : Type u_2} [Ring R] [AddCommGroup M] [Module R M] {ι : Type u_5} [Finite ι] [IsNoetherian R M] : IsNoetherian R (ι → M)

A version of isNoetherian_pi for non-dependent functions. We need this instance because sometimes Lean fails to apply the dependent version in non-dependent settings (e.g., it fails to prove that ι → ℝ is finite dimensional over ℝ).

instance isNoetherian_iSup {R : Type u_1} {P : Type u_4} [Ring R] [AddCommGroup P] [Module R P] {ι : Type u_5} [Finite ι] {M : ι → Submodule R P} [∀ (i : ι), IsNoetherian R ↥(M i)] : IsNoetherian R ↥(⨆ (i : ι), M i)

theorem isNoetherian_of_range_eq_ker {R : Type u_1} {M : Type u_2} {N : Type u_3} {P : Type u_4} [Ring R] [AddCommGroup M] [AddCommGroup N] [AddCommGroup P] [Module R M] [Module R N] [Module R P] [IsNoetherian R M] [IsNoetherian R P] (f : M →ₗ[R] N) (g : N →ₗ[R] P) (h : LinearMap.range f = LinearMap.ker g) : IsNoetherian R N

If the first and final modules in an exact sequence are Noetherian, then the middle module is also Noetherian.

theorem isNoetherian_iff_submodule_quotient {R : Type u_1} {P : Type u_4} [Ring R] [AddCommGroup P] [Module R P] (S : Submodule R P) : IsNoetherian R P ↔ IsNoetherian R ↥S ∧ IsNoetherian R (P ⧸ S)

instance isNoetherian_linearMap_pi (R : Type u_1) (M : Type u_2) [CommRing R] [AddCommGroup M] [Module R M] [IsNoetherian R M] {ι : Type u_4} [Finite ι] : IsNoetherian R ((ι → R) →ₗ[R] M)

instance isNoetherian_linearMap (R : Type u_1) (M : Type u_2) (N : Type u_3) [CommRing R] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] [IsNoetherian R M] [Module.Finite R N] : IsNoetherian R (N →ₗ[R] M)

theorem IsNoetherian.induction {R : Type u_1} {M : Type u_2} [Semiring R] [AddCommMonoid M] [Module R M] [IsNoetherian R M] {P : Submodule R M → Prop} (hgt : ∀ (I : Submodule R M), (∀ J > I, P J) → P I) (I : Submodule R M) : P I

If ∀ I > J, P I implies P J, then P holds for all submodules.

theorem LinearMap.isNoetherian_iff_of_bijective {R : Type u_1} {M : Type u_2} [Semiring R] [AddCommMonoid M] [Module R M] {S : Type u_3} {P : Type u_4} [Semiring S] [AddCommMonoid P] [Module S P] {σ : R →+* S} [RingHomSurjective σ] (l : M →ₛₗ[σ] P) (hl : Function.Bijective ⇑l) : IsNoetherian R M ↔ IsNoetherian S P

theorem Submodule.finite_ne_bot_of_iSupIndep {R : Type u_1} {M : Type u_2} [Semiring R] [AddCommMonoid M] [Module R M] [IsNoetherian R M] {ι : Type u_5} {N : ι → Submodule R M} (h : iSupIndep N) : {i : ι | N i ≠ ⊥}.Finite

theorem LinearIndependent.finite_of_isNoetherian {R : Type u_1} {M : Type u_2} [Semiring R] [AddCommMonoid M] [Module R M] [IsNoetherian R M] [Nontrivial R] {ι : Type u_5} {v : ι → M} (hv : LinearIndependent R v) : Finite ι

A linearly-independent family of vectors in a module over a non-trivial ring must be finite if the module is Noetherian.

theorem LinearIndependent.set_finite_of_isNoetherian {R : Type u_1} {M : Type u_2} [Semiring R] [AddCommMonoid M] [Module R M] [IsNoetherian R M] [Nontrivial R] {s : Set M} (hi : LinearIndependent R Subtype.val) : s.Finite

theorem IsNoetherian.disjoint_partialSups_eventually_bot {R : Type u_1} {M : Type u_2} [Semiring R] [AddCommMonoid M] [Module R M] [IsNoetherian R M] (f : ℕ → Submodule R M) (h : ∀ (n : ℕ), Disjoint ((partialSups f) n) (f (n + 1))) : ∃ (n : ℕ), ∀ (m : ℕ), n ≤ m → f m = ⊥

A sequence f of submodules of a noetherian module, with f (n+1) disjoint from the supremum of f 0, ..., f n, is eventually zero.

@[instance 100]

instance isNoetherian_of_subsingleton (R : Type u_1) (M : Type u_2) [Subsingleton R] [Semiring R] [AddCommMonoid M] [Module R M] : IsNoetherian R M

Modules over the trivial ring are Noetherian.

theorem isNoetherian_of_submodule_of_noetherian (R : Type u_1) (M : Type u_2) [Semiring R] [AddCommMonoid M] [Module R M] (N : Submodule R M) (h : IsNoetherian R M) : IsNoetherian R ↥N

theorem isNoetherian_of_tower (R : Type u_1) {S : Type u_2} {M : Type u_3} [Semiring R] [Semiring S] [AddCommMonoid M] [SMul R S] [Module S M] [Module R M] [IsScalarTower R S M] (h : IsNoetherian R M) : IsNoetherian S M

If M / S / R is a scalar tower, and M / R is Noetherian, then M / S is also noetherian.

instance isNoetherian_of_isNoetherianRing_of_finite (R : Type u_1) (M : Type u_2) [Ring R] [AddCommGroup M] [Module R M] [IsNoetherianRing R] [Module.Finite R M] : IsNoetherian R M

theorem isNoetherian_of_fg_of_noetherian {R : Type u_1} {M : Type u_2} [Ring R] [AddCommGroup M] [Module R M] (N : Submodule R M) [I : IsNoetherianRing R] (hN : N.FG) : IsNoetherian R ↥N

theorem isNoetherian_span_of_finite (R : Type u_1) {M : Type u_2} [Ring R] [AddCommGroup M] [Module R M] [IsNoetherianRing R] {A : Set M} (hA : A.Finite) : IsNoetherian R ↥(Submodule.span R A)

In a module over a Noetherian ring, the submodule generated by finitely many vectors is Noetherian.

theorem IsNoetherianRing.of_finite (R : Type u_1) (S : Type u_2) [Ring R] [Ring S] [Module R S] [IsScalarTower R S S] [IsNoetherianRing R] [Module.Finite R S] : IsNoetherianRing S

theorem isNoetherianRing_of_surjective (R : Type u_1) [Semiring R] (S : Type u_2) [Semiring S] (f : R →+* S) (hf : Function.Surjective ⇑f) [H : IsNoetherianRing R] : IsNoetherianRing S

instance isNoetherianRing_rangeS {R : Type u_1} [Semiring R] {S : Type u_2} [Semiring S] (f : R →+* S) [IsNoetherianRing R] : IsNoetherianRing ↥f.rangeS

instance isNoetherianRing_range {R : Type u_1} [Ring R] {S : Type u_2} [Ring S] (f : R →+* S) [IsNoetherianRing R] : IsNoetherianRing ↥f.range

theorem isNoetherianRing_of_ringEquiv (R : Type u_1) [Semiring R] {S : Type u_2} [Semiring S] (f : R ≃+* S) [IsNoetherianRing R] : IsNoetherianRing S

instance instIsNoetherianRingProd {R : Type u_1} {S : Type u_2} [Semiring R] [Semiring S] [IsNoetherianRing R] [IsNoetherianRing S] : IsNoetherianRing (R × S)

instance instIsNoetherianRingForallOfFinite {ι : Type u_2} [Finite ι] {R : ι → Type u_1} [(i : ι) → Semiring (R i)] [∀ (i : ι), IsNoetherianRing (R i)] : IsNoetherianRing ((i : ι) → R i)