Theorems · Theorem · commutative algebra
Ideal.comap_ne_bot_of_integral_mem
∀ {R : Type u_1} [inst : CommRing R] {S : Type u_2} [inst_1 : CommRing S] {I : Ideal S} [inst_2 : Algebra R S]
[Nontrivial R] [IsDomain S] {x : S}, x ≠ 0 → x ∈ I → IsIntegral R x → Ideal.comap (algebraMap R S) I ≠ ⊥- Defined in
- Mathlib.RingTheory.Ideal.GoingUp
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 120 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
- RingHomstatement · cited by 10,189
- Idealstatement and proof · cited by 4,748
- Bot.botstatement · cited by 4,720
- Algebra.algebraMapstatement · cited by 4,706
- Nontrivialstatement and proof · cited by 2,416
- IsDomainstatement and proof · cited by 2,196
- Ideal.comapstatement · cited by 443
- IsIntegralstatement and proof · cited by 427
- IsIntegral.isAlgebraicproof · cited by 18
- Ideal.comap_ne_bot_of_algebraic_memproof · cited by 1
Cited by2
Results whose statement or proof uses this declaration.
- Ideal.eq_bot_of_comap_eq_botproof · cited by 9
- Ideal.IsIntegral.comap_ne_botproof · cited by 4