lift_algebraMap — Mathlib

English

The image of lift is the subfield generated by the image of g, i.e., lift hg has fieldRange equal to Subfield.closure g.range.

Русский

Образ lift есть подтое поле, порождаемое образами g, то есть fieldRange( lift hg ) = Subfield.closure(g.range).

LaTeX

$$$\big( \mathrm{lift} \\! (hg) : K \to+* L \big).fieldRange = \operatorname{Subfield.closure} (g.range)$$$

Lean4

/-- Given a commutative ring `A` with field of fractions `K`,
and an injective ring hom `g : A →+* L` where `L` is a field,
the field hom induced from `K` to `L` maps `x` to `g x` for all
`x : A`. -/
@[simp]
theorem lift_algebraMap (hg : Injective g) (x) : lift hg (algebraMap A K x) = g x :=
  lift_eq _ _

Похожие утверждения