Theorems · Theorem · commutative algebra
Ideal.Quotient.nontrivial_of_liesOver_of_ne_top
∀ {A : Type u_3} {B : Type u_4} [inst : CommRing A] [inst_1 : CommRing B] [inst_2 : Algebra A B] (P : Ideal B)
{p : Ideal A} [P.LiesOver p], p ≠ ⊤ → Nontrivial (B ⧸ P)- Defined in
- Mathlib.RingTheory.Ideal.Over
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 72 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites9
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
- Top.topstatement and proof · cited by 9,680
- Idealstatement and proof · cited by 4,748
- Nontrivialstatement · cited by 2,416
- HasQuotient.Quotientstatement · cited by 2,301
- Ideal.LiesOverstatement and proof · cited by 272
- Ideal.Quotient.nontrivial_iffproof · cited by 9
- Ideal.ne_top_iff_of_liesOverproof · cited by 1
Cited by1
Results whose statement or proof uses this declaration.
- Ideal.Quotient.nontrivial_of_liesOver_of_isPrimeproof · cited by 1