Theorems · Theorem · nonassociative algebras
LieAlgebra.hasTrivialRadical_iff_no_solvable_ideals
∀ (R : Type u_1) (L : Type u_2) [inst : CommRing R] [inst_1 : LieRing L] [inst_2 : LieAlgebra R L], LieAlgebra.HasTrivialRadical R L ↔ ∀ (I : LieIdeal R L), LieAlgebra.IsSolvable ↥I → I = ⊥
- Defined in
- Mathlib.Algebra.Lie.Semisimple.Basic
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 81 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- CommRingLieRingLieAlgebra
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites9
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- CommRingstatement and proof · cited by 17,173
- Bot.botstatement · cited by 4,720
- LieRingstatement and proof · cited by 1,548
- LieAlgebrastatement and proof · cited by 1,246
- LieIdealstatement · cited by 282
- LieAlgebra.IsSolvablestatement · cited by 24
- LieAlgebra.HasTrivialRadicalstatement · cited by 11
- LieAlgebra.HasTrivialRadical.eq_bot_of_isSolvableproof · cited by 1
- LieAlgebra.hasTrivialRadical_of_no_solvable_idealsproof · cited by 1
Cited by1
Results whose statement or proof uses this declaration.
- LieAlgebra.hasTrivialRadical_iff_no_abelian_idealsproof · cited by 0