∀ {R : Type u_1} {S : Type u_2} [inst : CommRing R] [inst_1 : Ring S] [inst_2 : Algebra R S] [IsDomain R] [IsDomain S]
  [Module.Free R S] [Module.Finite R S] {x : S}, ((Algebra.norm R) x = 0) = (x = 0)