Theorems · Definition · commutative algebra
Ideal.primesOver
{A : Type u_2} →
[inst : CommSemiring A] → Ideal A → (B : Type u_3) → [inst_1 : Semiring B] → [Algebra A B] → Set (Ideal B)The set of all prime ideals in B that lie over an ideal p of A.
- Defined in
- Mathlib.RingTheory.Ideal.Over
- Cited by
- 84 results in Mathlib
- Foundations
- Depth 16 from the axioms · uses no axioms
- Assumes
- CommSemiringSemiringAlgebra
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.
- Setstatement · cited by 53,352
- Semiringstatement and proof · cited by 13,802
- Algebrastatement and proof · cited by 11,388
- CommSemiringstatement and proof · cited by 10,911
- Set.ofPredproof · cited by 6,101
- Idealstatement and proof · cited by 4,748
- Ideal.IsPrimeproof · cited by 827
- Ideal.LiesOverproof · cited by 272
Cited by90
Results whose statement or proof uses this declaration.
- NumberField.Ideal.primesOverSpanEquivMonicFactorsModstatement · cited by 10
- PrimeSpectrum.primesOverOrderIsoFiberstatement and proof · cited by 8
- Ideal.ncard_primesOver_mul_ramificationIdxIn_mul_inertiaDegInstatement and proof · cited by 5
- Ideal.ramificationIdx_posproof · cited by 5
- Ideal.fiberIsoOfBijectiveResidueFieldstatement · cited by 5
- IsDedekindDomain.coe_primesOverFinsetstatement and proof · cited by 4
- IsDedekindDomain.mem_primesOverFinset_iffstatement and proof · cited by 4
- IsDedekindDomain.primesOverEquivPrimesOverstatement and proof · cited by 4
- IsDedekindDomain.primesOver_ncard_ne_zerostatement · cited by 4
- IsCyclotomicExtension.Rat.eq_span_zeta_sub_one_of_liesOverproof · cited by 3
- Ideal.ne_bot_of_mem_primesOverstatement and proof · cited by 3
- Ideal.primesOver.mkstatement · cited by 3