Theorems · Theorem · commutative algebra
Ideal.exists_maximal_ideal_liesOver_of_isIntegral
∀ {R : Type u_1} [inst : CommRing R] {S : Type u_2} [inst_1 : CommRing S] [inst_2 : Algebra R S]
[Algebra.IsIntegral R S] [FaithfulSMul R S] (P : Ideal R) [P.IsMaximal], ∃ Q, Q.IsMaximal ∧ Q.LiesOver P- Defined in
- Mathlib.RingTheory.Ideal.GoingUp
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 123 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites12
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
- Algebrastatement and proof · cited by 11,388
- Idealstatement and proof · cited by 4,748
- Algebra.algebraMapproof · cited by 4,706
- Ideal.IsMaximalstatement and proof · cited by 452
- FaithfulSMulstatement and proof · cited by 340
- Ideal.LiesOverstatement · cited by 272
- Algebra.IsIntegralstatement and proof · cited by 224
- FaithfulSMul.algebraMap_injectiveproof · cited by 198
- Ideal.underproof · cited by 170
- RingHom.injective_iff_ker_eq_botproof · cited by 29
- Ideal.exists_ideal_over_maximal_of_isIntegralproof · cited by 5
Cited by3
Results whose statement or proof uses this declaration.
- IsDedekindDomain.primesOver_ncard_ne_zeroproof · cited by 4
- IsLocalRing.primesOver_eqproof · cited by 1
- Ideal.relNorm_eq_pow_of_isMaximalproof · cited by 1