Theorems · Theorem · commutative algebra
Ideal.Quotient.maximal_of_isField
∀ {R : Type u_5} [inst : CommRing R] (I : Ideal R), IsField (R ⧸ I) → I.IsMaximalIf the quotient by an ideal is a field, then the ideal is maximal.
- Defined in
- Mathlib.RingTheory.Ideal.Quotient.Basic
- Cited by
- 4 results in Mathlib
- Foundations
- Depth 84 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- CommRing
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites19
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coeproof · cited by 62,936
- CommRingstatement and proof · cited by 17,173
- Idealstatement and proof · cited by 4,748
- mul_oneproof · cited by 3,885
- zero_addproof · cited by 2,366
- HasQuotient.Quotientstatement and proof · cited by 2,301
- sub_selfproof · cited by 996
- Ideal.Quotient.mkproof · cited by 610
- Ideal.IsMaximalstatement · cited by 452
- Ideal.mul_mem_leftproof · cited by 107
- IsFieldstatement and proof · cited by 103
- Ideal.Quotient.eq_zero_iff_memproof · cited by 74
Cited by4
Results whose statement or proof uses this declaration.
- Ideal.Quotient.maximal_ideal_iff_isField_quotientproof · cited by 7
- Ideal.isMaximal_comap_of_isIntegral_of_isMaximalproof · cited by 3
- IsArithFrobAt.exists_of_isInvariantproof · cited by 1
- Ring.HasFiniteQuotients.maximalOfPrimeproof · cited by 0