Theorems · Definition · field theory

RatFunc.mapAlgHom

{K : Type u} →
  [inst : CommRing K] →
    [inst_1 : IsDomain K] →
      {R : Type u_2} →
        {S : Type u_3} →
          [inst_2 : CommRing R] →
            [inst_3 : IsDomain R] →
              [inst_4 : CommSemiring S] →
                [inst_5 : Algebra S (Polynomial K)] →
                  [inst_6 : Algebra S (Polynomial R)] →
                    (φ : Polynomial K →ₐ[S] Polynomial R) →
                      nonZeroDivisors (Polynomial K) ≤ Submonoid.comap φ (nonZeroDivisors (Polynomial R)) →
                        RatFunc K →ₐ[S] RatFunc R

Lift an algebra homomorphism that maps polynomials φ : K[X] →ₐ[S] R[X] to a K⟮X⟯ →ₐ[S] R⟮X⟯, on the condition that φ maps non-zero-divisors to non-zero-divisors, by mapping both the numerator and denominator and quotienting them.

Defined in

Mathlib.FieldTheory.RatFunc.Basic

Cited by

1 results in Mathlib

Foundations

Depth 120 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingIsDomainCommRingIsDomainCommSemiringAlgebraAlgebra