imc_equality_template
In a commutative ring, mutual divisibility of two elements forces the principal ideals they generate to coincide. Researchers on the Birch-Swinnerton-Dyer conjecture cite this as the algebraic core of BSD Stage 6. The proof is a direct one-line application of the mutual-divisibility lemma for principal ideals.
claimIn a commutative ring $R$, if $A$ divides $B$ and $B$ divides $A$, then the principal ideal generated by $A$ equals the principal ideal generated by $B$.
background
The Pinch Algebra module develops tools for mutual divisibility in commutative rings, targeting UFDs and Fredholm obstructions as part of Foundation paper F5. Key prior result is the lemma that in any commutative ring, the principal ideals $(a)$ and $(b)$ coincide precisely when $a$ divides $b$ and $b$ divides $a$ (quoted from upstream: 'In a commutative ring, (a) = (b) iff a | b and b | a').
proof idea
This is a one-line wrapper that applies the principal_ideal_eq_of_mutual_dvd lemma directly to the two divisibility hypotheses.
why it matters in Recognition Science
This supplies the algebraic heart of BSD Stage 6 inside the Recognition Science framework. It occupies the F5.2.3 slot in the Pinch Algebra section of Foundation paper F5, which treats mutual divisibility in UFDs and Fredholm obstructions. The result supports primary BSD applications and secondary Yang-Mills work; the proof closes the template with no remaining scaffolding.
scope and limits
- Does not assume the ring is a UFD or integral domain.
- Does not extend equality to non-principal ideals.
- Does not address non-commutative rings.
- Does not supply norm bounds or generator counts.
formal statement (Lean)
47theorem imc_equality_template {R : Type*} [CommRing R]
48 {A B : R} (hAB : A ∣ B) (hBA : B ∣ A) :
49 Ideal.span ({A} : Set R) = Ideal.span {B} :=
proof body
Term-mode proof.
50 principal_ideal_eq_of_mutual_dvd hAB hBA
51
52/-! ## §3. Fredholm / operator pinch template -/
53
54/-- **F5.3.2**: A function that is not surjective cannot map finite sets
55 onto infinite sets. (Basic set-theoretic obstruction.)
56 This is the finite-capacity veto in its simplest form. -/