English
In a Noetherian environment, any property P of subalgebras that holds for ⊥ and is preserved when adjoining a single element to the algebra, holds for all subalgebras.
Русский
В окружении Нётера, если P верно для ⊥ и сохраняется при adjoin одного элемента, тогда P верно для всей подалгебры.
LaTeX
$$$\\text{If } P(\\bot) \\text{ and } (\\forall S,x, P(S) \\Rightarrow P(\\operatorname{adjoin}_R (insert x S))) \\Rightarrow \\forall S, P(S)$$$
Lean4
theorem prod {S : Subalgebra R A} {T : Subalgebra R B} (hS : S.FG) (hT : T.FG) : (S.prod T).FG :=
by
obtain ⟨s, hs⟩ := fg_def.1 hS
obtain ⟨t, ht⟩ := fg_def.1 hT
rw [← hs.2, ← ht.2]
exact
fg_def.2
⟨LinearMap.inl R A B '' (s ∪ { 1 }) ∪ LinearMap.inr R A B '' (t ∪ { 1 }),
Set.Finite.union (Set.Finite.image _ (Set.Finite.union hs.1 (Set.finite_singleton _)))
(Set.Finite.image _ (Set.Finite.union ht.1 (Set.finite_singleton _))),
Algebra.adjoin_inl_union_inr_eq_prod R s t⟩
