Theorems · Theorem · commutative algebra
IsPrincipalIdealRing.of_prime
∀ {R : Type u_1} [inst : CommSemiring R],
(∀ (P : Ideal R), P.IsPrime → Submodule.IsPrincipal P) → IsPrincipalIdealRing RIf all prime ideals in a commutative ring are principal, so are all other ideals.
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 79 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.
Cites9
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- CommSemiringstatement and proof · cited by 10,911
- Idealstatement and proof · cited by 4,748
- Ideal.IsPrimestatement and proof · cited by 827
- IsPrincipalIdealRingstatement · cited by 131
- Submodule.IsPrincipalstatement and proof · cited by 129
- Ideal.IsOka.forall_of_forall_primeproof · cited by 2
- Ideal.isOka_isPrincipalproof · cited by 1
- Ideal.exists_maximal_not_isPrincipalproof · cited by 1
- isPrincipalIdealRing_iffproof · cited by 1
Cited by1
Results whose statement or proof uses this declaration.
- IsPrincipalIdealRing.of_prime_ne_botproof · cited by 1