Theorems · Definition · ring theory

NonUnitalStarSubalgebra.unitizationStarAlgEquiv

{R : Type u_1} →
  {S : Type u_2} →
    {A : Type u_3} →
      [inst : Field R] →
        [inst_1 : StarRing R] →
          [inst_2 : Ring A] →
            [inst_3 : StarRing A] →
              [inst_4 : Algebra R A] →
                [inst_5 : StarModule R A] →
                  [inst_6 : SetLike S A] →
                    [hSA : NonUnitalSubringClass S A] →
                      [hSRA : SMulMemClass S R A] →
                        [inst_7 : StarMemClass S A] →
                          (s : S) → 1 ∉ s → Unitization R ↥s ≃⋆ₐ[R] ↥(StarAlgebra.adjoin R ↑s)

If a NonUnitalStarSubalgebra over a field does not contain 1, then its unitization is isomorphic to its StarAlgebra.adjoin.

Defined in

Mathlib.Algebra.Algebra.Subalgebra.Unitization

Cited by

1 results in Mathlib

Foundations

Depth 80 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

FieldStarRingRingStarRingAlgebraStarModuleSetLikeNonUnitalSubringClassSMulMemClassStarMemClass