Jacobson.lean`

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Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

4 declarations, 3 contain sorry, 0 declared as axioms

Jacobson Conjecture

References:

Wikipedia
[He1965]: Herstein, I. N. (1965), "A counterexample in Noetherian rings", Proceedings of the National Academy of Sciences of the United States of America, 54 (4): 1036–1037 https://pmc.ncbi.nlm.nih.gov/articles/PMC219788/
[Ja1956]: Jacobson, Nathan. Structure of rings. Vol. 37. American Mathematical Soc., 1956.
[Le1977]: Lenagan, T. H. (1977), "Noetherian rings with Krull dimension one", J. London Math. Soc. Series 2, 15 (1): 41–47 https://londmathsoc.onlinelibrary.wiley.com/doi/abs/10.1112/jlms/s2-15.1.41

Imports (1)

FormalConjecturesUtil

def Jacobson.JacobsonConjectureFor The Jacobson conjecture for a ring $R$: the infimum of the powers of the Jacobson radical $J$ is the zero ideal line 44 source
Source documentation
The Jacobson conjecture for a ring $R$: the infimum of the powers of the Jacobson radical $J$ is the zero ideal.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Ring
Proof uses: Eq, Ideal, iInf, Nat, HPow.hPow, Ring.jacobson, OfNat.ofNat
9 instances, coercions and auxiliary constants left out.
```
def JacobsonConjectureFor (R : Type u) [Ring R] : Prop :=
…
```
theorem Jacobson.jacobson_conjecture The Jacobson conjecture (in its modern form): In a (noncommutative) ring which is left and right Noetherian, the intersection of the powers of the Jacobson ideal is trivial sorry line 51 source
Source documentation
The Jacobson conjecture (in its modern form): In a (noncommutative) ring which is left and right Noetherian, the intersection of the powers of the Jacobson ideal is trivial

depends on sorry what this means
Statement uses: Iff, True, Ring, IsNoetherianRing, IsRightNoetherianRing, Jacobson.JacobsonConjectureFor
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Nat
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem jacobson_conjecture :
    answer(sorry) ↔ ∀ (R : Type) [Ring R] [IsNoetherianRing R] [IsRightNoetherianRing R],
      JacobsonConjectureFor R := by
…
```
Mentions: IsRightNoetherianRing, Jacobson.JacobsonConjectureFor
theorem Jacobson.jacobson_conjecture_of_comm_ring For commutative rings this is the case as a consequence of Krull's intersection theorem sorry line 58 source
Source documentation
For commutative rings this is the case as a consequence of Krull's intersection theorem.

depends on sorry what this means
Statement uses: CommRing, IsNoetherianRing, Jacobson.JacobsonConjectureFor
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Nat
7 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem jacobson_conjecture_of_comm_ring (R : Type u) [CommRing R] [IsNoetherianRing R] :
    JacobsonConjectureFor R := by
…
```
Mentions: Jacobson.JacobsonConjectureFor
theorem Jacobson.jacobson_conjecture_of_right_noetherian Originally, on page 200 of [Ja1956], Jacobson asked if the Jacobson conjecture holds for all right Noetherian rings sorry line 66 source
Source documentation
Originally, on page 200 of [Ja1956], Jacobson asked if the Jacobson conjecture holds for all right Noetherian rings. However in [He1965] Herstein constructs a right Noetherian ring for which the Jacobson conjecture does not hold.

depends on sorry what this means
Statement uses: Iff, False, Ring, IsRightNoetherianRing, Jacobson.JacobsonConjectureFor
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Nat
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem jacobson_conjecture_of_right_noetherian :
    answer(False) ↔ ∀ (R : Type) [Ring R] [IsRightNoetherianRing R], JacobsonConjectureFor R := by
…
```
Mentions: IsRightNoetherianRing, Jacobson.JacobsonConjectureFor