Theorems · Definition · linear algebra

Submodule.dualQuotEquivDualAnnihilator

{R : Type u_1} →
  {M : Type u_2} →
    [inst : CommRing R] →
      [inst_1 : AddCommGroup M] →
        [inst_2 : Module R M] → (W : Submodule R M) → Module.Dual R (M ⧸ W) ≃ₗ[R] ↥W.dualAnnihilator

Equivalence $(M/W)^* \cong \operatorname{ann}(W)$. That is, there is a one-to-one correspondence between the dual of M ⧸ W and those elements of the dual of M that vanish on W. The inverse of this is Submodule.dualCopairing.

Defined in

Mathlib.LinearAlgebra.Dual.Lemmas

Cited by

9 results in Mathlib

Foundations

Depth 88 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingAddCommGroupModule