Theorems · Definition · commutative algebra

LinearMap.ringLmapEquivSelf

(R : Type u_1) →
  (S : Type u_4) →
    (M : Type u_5) →
      [inst : Semiring R] →
        [inst_1 : Semiring S] →
          [inst_2 : AddCommMonoid M] →
            [inst_3 : Module R M] → [inst_4 : Module S M] → [inst_5 : SMulCommClass R S M] → (R →ₗ[R] M) ≃ₗ[S] M

The equivalence between R-linear maps from R to M, and points of M itself. This says that the forgetful functor from R-modules to types is representable, by R. This is an S-linear equivalence, under the assumption that S acts on M commuting with R. When R is commutative, we can take this to be the usual action with S = R. Otherwise, S = ℕ shows that the equivalence is additive. See note [bundled maps over different rings].

Defined in

Mathlib.Algebra.Module.Equiv.Basic

Cited by

12 results in Mathlib

Foundations

Depth 35 from the axioms · uses propext, Quot.sound

Assumes

SemiringSemiringAddCommMonoidModuleModuleSMulCommClass