Theorems · Theorem · commutative algebra

LinearMap.range_le_ker_iff

∀ {R : Type u_1} {R₂ : Type u_2} {R₃ : Type u_3} {M : Type u_5} {M₂ : Type u_6} {M₃ : Type u_7} [inst : Semiring R]
  [inst_1 : Semiring R₂] [inst_2 : Semiring R₃] [inst_3 : AddCommMonoid M] [inst_4 : AddCommMonoid M₂]
  [inst_5 : AddCommMonoid M₃] [inst_6 : Module R M] [inst_7 : Module R₂ M₂] [inst_8 : Module R₃ M₃] {τ₁₂ : R →+* R₂}
  {τ₂₃ : R₂ →+* R₃} {τ₁₃ : R →+* R₃} [inst_9 : RingHomCompTriple τ₁₂ τ₂₃ τ₁₃] [inst_10 : RingHomSurjective τ₁₂]
  {f : M →ₛₗ[τ₁₂] M₂} {g : M₂ →ₛₗ[τ₂₃] M₃}, f.range ≤ g.ker ↔ g ∘ₛₗ f = 0

Defined in

Mathlib.Algebra.Module.Submodule.Range

Cited by

9 results in Mathlib

Foundations

Depth 30 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

SemiringSemiringSemiringAddCommMonoidAddCommMonoidAddCommMonoidModuleModuleModuleRingHomCompTripleRingHomSurjective