Theorems · Theorem · commutative algebra
Ideal.eq_bot_or_top
∀ {K : Type u} [inst : DivisionSemiring K] (I : Ideal K), I = ⊥ ∨ I = ⊤All ideals in a division (semi)ring are trivial.
- Defined in
- Mathlib.RingTheory.Ideal.Lattice
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 28 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- DivisionSemiring
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites8
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Top.topstatement · cited by 9,680
- Idealstatement and proof · cited by 4,748
- Bot.botstatement and proof · cited by 4,720
- inv_mul_cancel₀proof · cited by 267
- DivisionSemiringstatement and proof · cited by 216
- eq_bot_iffproof · cited by 159
- Ideal.mul_mem_leftproof · cited by 107
- Ideal.ne_top_iff_oneproof · cited by 16
Cited by2
Results whose statement or proof uses this declaration.
- Ideal.bot_isMaximalproof · cited by 5
- Ideal.eq_bot_of_primeproof · cited by 2