Theorems · Theorem · commutative algebra
Ideal.Quotient.isDomain_iff_prime
∀ {R : Type u_3} [inst : Ring R] (I : Ideal R) [inst_1 : I.IsTwoSided], IsDomain (R ⧸ I) ↔ I.IsPrime- Defined in
- Mathlib.RingTheory.Ideal.Quotient.Basic
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 86 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- RingIdeal.IsTwoSided
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites14
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Ringstatement and proof · cited by 7,463
- Idealstatement and proof · cited by 4,748
- Nontrivialproof · cited by 2,416
- HasQuotient.Quotientstatement and proof · cited by 2,301
- IsDomainstatement and proof · cited by 2,196
- Ideal.IsPrimestatement and proof · cited by 827
- Ideal.Quotient.mkproof · cited by 610
- NoZeroDivisorsproof · cited by 545
- Ideal.IsTwoSidedstatement and proof · cited by 179
- zero_ne_oneproof · cited by 90
- RingHom.map_mulproof · cited by 45
- NoZeroDivisors.eq_zero_or_eq_zero_of_mul_eq_zeroproof · cited by 14
Cited by1
Results whose statement or proof uses this declaration.
- AdjoinRoot.isDomain_of_primeproof · cited by 1