Theorems · Theorem · commutative algebra
Submodule.IsPrincipal.prime_generator_of_isPrime
∀ {R : Type u} [inst : CommSemiring R] (S : Ideal R) [inst_1 : Submodule.IsPrincipal S] [is_prime : S.IsPrime],
S ≠ ⊥ → Prime (Submodule.IsPrincipal.generator S)- Defined in
- Mathlib.RingTheory.PrincipalIdealDomain
- Cited by
- 4 results in Mathlib
- Foundations
- Depth 30 from the axioms · uses propext, Classical.choice, Quot.sound
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.
- CommSemiringstatement and proof · cited by 10,911
- Idealstatement and proof · cited by 4,748
- Bot.botstatement and proof · cited by 4,720
- IsUnitproof · cited by 1,602
- Ideal.IsPrimestatement and proof · cited by 827
- Primestatement · cited by 277
- Submodule.IsPrincipalstatement and proof · cited by 129
- Ideal.IsPrime.ne_topproof · cited by 82
- Submodule.IsPrincipal.generatorstatement and proof · cited by 56
- Ideal.eq_top_of_isUnit_memproof · cited by 26
- Submodule.IsPrincipal.mem_iff_generator_dvdproof · cited by 11
- Submodule.IsPrincipal.generator_memproof · cited by 9
Cited by4
Results whose statement or proof uses this declaration.
- UniqueFactorizationMonoid.of_forall_isPrincipal_of_height_eq_oneproof · cited by 1
- Rat.HeightOneSpectrum.prime_natGeneratorproof · cited by 1
- Submodule.exists_isInternal_prime_power_torsion_of_pidproof · cited by 1
- Ideal.prime_generator_of_primeproof · cited by 0