Theorems · Theorem · commutative algebra

Ring.DirectLimit.of_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) [IsDirectedOrder ι] [DirectedSystem G fun i j h => ⇑(f' i j h)],
  (∀ (i j : ι) (hij : i ≤ j), Function.Injective ⇑(f' i j hij)) →
    ∀ (i : ι), Function.Injective ⇑(Ring.DirectLimit.of G (fun i j h => ⇑(f' i j h)) i)

If the maps in the directed system are injective, then the canonical maps from the components to the direct limits are injective.

Defined in

Mathlib.Algebra.Colimit.Ring

Cited by

0 results in Mathlib

Foundations

Depth 118 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

PreorderCommRingIsDirectedOrderDirectedSystem