MathOverflowCommutative algebra
Mathoverflow 507128
There exists a proper ideal I in a (commutative) total ring R of fractions that is an invertible module. If I ⊊ R is such an example, I must have infinite order in the Picard group, and R must not be Noetherian (otherwise it must be semi-local and...
Mathematical statement
There exists a proper ideal I in a (commutative) total ring R of fractions that is an
invertible module. If I ⊊ R is such an example, I must have infinite order in the Picard group,
and R must not be Noetherian (otherwise it must be semi-local and therefore have trivial Picard
group).
Statement source: MathOverflow statement material
Statement terms: CC-BY-SA-4.0
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Statement artifacts, not proofs
These records expose exact Lean propositions and statement-only wrappers. Defining a proposition does not supply a proof of it. A placeholder-bearing target also contains no proof. Elaboration checks syntax and types; it does not certify that a formalization perfectly captures every nuance of the informal problem.
Pinned Lean formulation 1
exists_isFractionRing_self_ideal_ne_top_invertible
theorem exists_isFractionRing_self_ideal_ne_top_invertible : ∃ (R : Type) (_ : CommRing R) (_ : IsFractionRing R R) (I : Ideal R), I ≠ ⊤ ∧ Module.Invertible R I := by sorry- Statement source
- Formal Conjectures
- Lean version
- v4.27.0
- Placeholder
- Present; no proof artifact
- Source evidence
- Pinned source index
- Fidelity review
- Community formulation
References