Theorems · Theorem · commutative algebra

Ring.DirectLimit.exists_of

∀ {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} [Nonempty ι] [IsDirectedOrder ι] (z : Ring.DirectLimit G f),
  ∃ i x, (Ring.DirectLimit.of G f i) x = z

Every element of the direct limit corresponds to some element in some component of the directed system.

Defined in

Mathlib.Algebra.Colimit.Ring

Cited by

2 results in Mathlib

Foundations

Depth 114 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

PreorderCommRingNonemptyIsDirectedOrder