Theorems · Theorem · commutative algebra
Ideal.isPrime_of_maximally_disjoint
∀ {α : Type u} [inst : CommSemiring α] (I : Ideal α) (S : Submonoid α),
Disjoint ↑I ↑S → (∀ (J : Ideal α), I < J → ¬Disjoint ↑J ↑S) → I.IsPrime- Defined in
- Mathlib.RingTheory.Ideal.Maximal
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 67 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- CommSemiring
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites16
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement · cited by 53,352
- CommSemiringstatement and proof · cited by 10,911
- Top.topproof · cited by 9,680
- SetLike.coestatement and proof · cited by 8,199
- Idealstatement and proof · cited by 4,748
- Submonoidstatement and proof · cited by 3,086
- Disjointstatement and proof · cited by 2,201
- Ideal.spanproof · cited by 948
- Ideal.IsPrimestatement · cited by 827
- Ideal.mul_mem_leftproof · cited by 107
- Ideal.mul_mem_rightproof · cited by 71
- Ideal.add_memproof · cited by 36
Cited by1
Results whose statement or proof uses this declaration.
- Ideal.exists_le_prime_disjointproof · cited by 5