Theorems · Theorem · commutative algebra
Ring.isField_iff_maximal_bot
∀ {R : Type u_5} [inst : CommSemiring R] [Nontrivial R], IsField R ↔ ⊥.IsMaximal- Defined in
- Mathlib.RingTheory.Ideal.Basic
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 46 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- CommSemiringNontrivial
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites15
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- CommSemiringstatement and proof · cited by 10,911
- Idealstatement · cited by 4,748
- Bot.botstatement and proof · cited by 4,720
- Nontrivialstatement and proof · cited by 2,416
- IsUnitproof · cited by 1,602
- Ideal.IsMaximalstatement and proof · cited by 452
- Semifieldproof · cited by 439
- bot_leproof · cited by 306
- IsFieldstatement and proof · cited by 103
- Ideal.IsMaximal.eq_of_leproof · cited by 39
- Ideal.span_singleton_eq_botproof · cited by 27
- Ideal.bot_isMaximalproof · cited by 5
Cited by2
Results whose statement or proof uses this declaration.
- Ring.exists_maximal_of_not_isFieldproof · cited by 2
- Ring.not_isField_of_ne_of_neproof · cited by 2