English
Let R be a commutative semiring, A a semiring with R-algebra structure, and M an additive monoid with an A-module structure, with IsScalarTower. If there exists a polynomial p ∈ R[X] such that aeval(a) p = 0 and p ≠ 0, then AEval R M a is a torsion module over R[X]; i.e., every element is annihilated by some nonzero polynomial.
Русский
Пусть R — коммутативное полиномиальное кольцо, A — полугруппа с структурой R-алгебры, M — аддитивная моноида с структурой модуля над A и IsScalarTower. Если существует p ∈ R[X] такая, что aeval(a) p = 0 и p ≠ 0, тогда AEval R M a является торовым модулем над R[X].
LaTeX
$$$\exists p \in R[X],\; p \neq 0 \text{ и } aeval\ a\ p = 0 \quad \Longrightarrow\quad \text{IsTorsion } R[X] (AEval\ R\ M\ a).$$$
Lean4
theorem isTorsion_of_aeval_eq_zero [CommSemiring R] [NoZeroDivisors R] [Semiring A] [Algebra R A] [AddCommMonoid M]
[Module A M] [Module R M] [IsScalarTower R A M] {p : R[X]} (h : aeval a p = 0) (h' : p ≠ 0) :
IsTorsion R[X] (AEval R M a) :=
by
have hp : p ∈ nonZeroDivisors R[X] :=
mem_nonZeroDivisors_iff_right.mpr fun q hq ↦ Or.resolve_right (mul_eq_zero.mp hq) h'
exact fun x ↦ ⟨⟨p, hp⟩, (of R M a).symm.injective <| by simp [h]⟩
