Theorems · Theorem · ring theory
IsSimpleRing.tfae
∀ {R : Type u} [inst : Ring R] [IsSimpleRing R], [IsSemisimpleRing R, IsArtinianRing R, ∃ I, IsAtom I].TFAEA simple ring is semisimple iff it is Artinian, iff it has a minimal left ideal.
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 96 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- RingIsSimpleRing
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites28
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Top.topproof · cited by 9,680
- Ringstatement and proof · cited by 7,463
- Idealstatement and proof · cited by 4,748
- Bot.botproof · cited by 4,720
- Nontrivialproof · cited by 2,416
- LinearEquiv.symmproof · cited by 1,461
- LE.le.trans_eqproof · cited by 328
- LT.lt.not_geproof · cited by 305
- LinearEquiv.transproof · cited by 298
- Ideal.IsTwoSidedproof · cited by 179
- LinearEquiv.reflproof · cited by 143
- IsAtomstatement and proof · cited by 130
Cited by1
Results whose statement or proof uses this declaration.
- IsSimpleRing.isSemisimpleRing_iff_isArtinianRingproof · cited by 1