Theorems · Theorem · ring theory

TrivSqZeroExt.range_liftAux

∀ {S : Type u_1} {R : Type u} {M : Type v} [inst : CommSemiring S] [inst_1 : Semiring R] [inst_2 : AddCommMonoid M]
  [inst_3 : Algebra S R] [inst_4 : Module S M] [inst_5 : Module R M] [inst_6 : Module Rᵐᵒᵖ M]
  [inst_7 : SMulCommClass R Rᵐᵒᵖ M] [inst_8 : IsScalarTower S R M] [inst_9 : IsScalarTower S Rᵐᵒᵖ M] {A : Type u_2}
  [inst_10 : Semiring A] [inst_11 : Algebra S A] (f : R →ₐ[S] A) (g : M →ₗ[S] A) (hg : ∀ (x y : M), g x * g y = 0)
  (hfg : ∀ (r : R) (x : M), g (r • x) = f r * g x) (hgf : ∀ (r : R) (x : M), g (MulOpposite.op r • x) = g x * f r),
  (TrivSqZeroExt.lift f g hg hfg hgf).range = f.range ⊔ Algebra.adjoin S (Set.range ⇑g)

Defined in

Mathlib.Algebra.TrivSqZeroExt.Basic

Cited by

0 results in Mathlib

Foundations

Depth 77 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommSemiringSemiringAddCommMonoidAlgebraModuleModuleModuleSMulCommClassIsScalarTowerIsScalarTowerSemiringAlgebra