English
Let f: A →ₐ[R] B be an algebra hom and S ⊆ A a subalgebra. Then (S.map f).comap f equals S joined with the ℝ-adjoin of f^{-1}({0}).
Русский
Пусть f: A →ₐ[R] B — алгебра-гомоморфизм, S ⊆ A — подалгебра. Тогда (S.map f).comap f равно S объединённой с ℝ-адъюнктом f^{-1}({0}).
LaTeX
$$$ (\operatorname{Subalgebra.map} f\ S).\operatorname{comap} f = S \;⊔\; \operatorname{Algebra.adjoin}_{R}(f^{-1}(\{0\})) $$$
Lean4
/-- A version of `Set.matrix` for `Subalgebra`s.
Given a `Subalgebra` `S`, `S.matrix` is the `Subalgebra` of square matrices `m`
all of whose entries `m i j` belong to `S`. -/
@[simps!]
def matrix (S : Subalgebra R A) : Subalgebra R (Matrix n n A)
where
__ := S.toSubsemiring.matrix
algebraMap_mem' _ := (diagonal_mem_matrix_iff (Subalgebra.zero_mem _)).mpr (fun _ => Subalgebra.algebraMap_mem _ _)
