Theorems · Theorem · commutative algebra

Ring.DirectLimit.lift_injective

∀ {ι : Type u_1} [inst : Preorder ι] {G : ι → Type u_2} [inst_1 : (i : ι) → CommRing (G i)]
  {f : (i j : ι) → i ≤ j → G i → G j} (P : Type u_3) [inst_2 : CommRing P] (g : (i : ι) → G i →+* P)
  (Hg : ∀ (i j : ι) (hij : i ≤ j) (x : G i), (g j) (f i j hij x) = (g i) x) [Nonempty ι] [IsDirectedOrder ι],
  (∀ (i : ι), Function.Injective ⇑(g i)) → Function.Injective ⇑(Ring.DirectLimit.lift G f P g Hg)

Defined in

Mathlib.Algebra.Colimit.Ring

Cited by

0 results in Mathlib

Foundations

Depth 116 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

PreorderCommRingCommRingNonemptyIsDirectedOrder