Theorems · Theorem · commutative algebra
Ideal.IsPrime.isMaximal
∀ {R : Type u_4} [inst : CommRing R] [Ring.DimensionLEOne R] {p : Ideal R}, p.IsPrime → p ≠ ⊥ → p.IsMaximal- Defined in
- Mathlib.RingTheory.DedekindDomain.Basic
- Cited by
- 24 results in Mathlib
- Foundations
- Depth 17 from the axioms · uses propext, Quot.sound
- Assumes
- CommRingRing.DimensionLEOne
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites7
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
- Idealstatement and proof · cited by 4,748
- Bot.botstatement and proof · cited by 4,720
- Ideal.IsPrimestatement and proof · cited by 827
- Ideal.IsMaximalstatement · cited by 452
- Ring.DimensionLEOnestatement and proof · cited by 12
- Ring.DimensionLEOne.maximalOfPrimeproof · cited by 6
Cited by24
Results whose statement or proof uses this declaration.
- Ideal.eq_prime_pow_mul_coprimeproof · cited by 3
- IsDedekindDomain.inf_pow_eq_prod_of_primeproof · cited by 3
- Ideal.ramificationIdx'_eq_ramificationIdx'proof · cited by 3
- Ideal.absNorm_eq_pow_inertiaDeg'_of_liesOverproof · cited by 2
- Ring.DimensionLEOne.not_lt_ltproof · cited by 2
- Ring.DimensionLEOne.of_isIntegralproof · cited by 2
- Ideal.exists_isMaximal_dvd_of_dvd_absNormproof · cited by 2
- PrimeSpectrum.exists_multiset_prod_cons_le_and_prod_not_leproof · cited by 2
- IsDiscreteValuationRing.ringKrullDim_eq_oneproof · cited by 1
- maximalIdeal_isPrincipal_of_isDedekindDomainproof · cited by 1
- Ring.DimensionLeOne.prime_le_prime_iff_eqproof · cited by 1
- not_dvd_differentIdeal_iffproof · cited by 1