Theorems · Theorem · commutative algebra

Submodule.fg_map_iff

∀ {R : Type u_1} {M : Type u_2} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {S : Type u_3}
  {P : Type u_4} [inst_3 : Semiring S] [inst_4 : AddCommMonoid P] [inst_5 : Module S P] {σ : R →+* S}
  [inst_6 : RingHomSurjective σ] (f : M →ₛₗ[σ] P),
  Function.Injective ⇑f → ∀ {N : Submodule R M}, (Submodule.map f N).FG ↔ N.FG

Defined in

Mathlib.RingTheory.Finiteness.Basic

Cited by

1 results in Mathlib

Foundations

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

Assumes

SemiringAddCommMonoidModuleSemiringAddCommMonoidModuleRingHomSurjective