Theorems · Theorem · commutative algebra

Submodule.fg_of_fg_map_of_fg_inf_ker

∀ {R : Type u_1} {M : Type u_2} {P : Type u_4} [inst : Ring R] [inst_1 : AddCommGroup M] [inst_2 : Module R M]
  [inst_3 : AddCommGroup P] [inst_4 : Module R P] (f : M →ₗ[R] P) {s : Submodule R M},
  (Submodule.map f s).FG → (s ⊓ f.ker).FG → s.FG

If 0 → M' → M → M'' → 0 is exact and M' and M'' are finitely generated then so is M.

Defined in

Mathlib.RingTheory.Finiteness.Finsupp

Cited by

7 results in Mathlib

Foundations

Depth 81 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RingAddCommGroupModuleAddCommGroupModule