Theorems · Theorem · commutative algebra
Ideal.span_singleton_prime
∀ {α : Type u} [inst : CommSemiring α] {p : α}, p ≠ 0 → ((Ideal.span {p}).IsPrime ↔ Prime p)- Defined in
- Mathlib.RingTheory.Ideal.Maximal
- Cited by
- 27 results in Mathlib
- Foundations
- Depth 32 from the axioms · uses propext, Quot.sound
- Assumes
- CommSemiring
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites6
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
- IsUnitproof · cited by 1,602
- Ideal.spanstatement · cited by 948
- Ideal.IsPrimestatement · cited by 827
- Primestatement · cited by 277
Cited by27
Results whose statement or proof uses this declaration.
- Ideal.prime_span_singleton_iffproof · cited by 5
- PowerSeries.X_primeproof · cited by 3
- Ideal.eq_span_singleton_of_height_eq_oneproof · cited by 2
- PowerSeries.intValuation_eq_of_coeproof · cited by 2
- Prime.isMaximal_span_singletonproof · cited by 2
- IsDiscreteValuationRing.iff_pid_with_one_nonzero_primeproof · cited by 2
- Ideal.count_associates_eqproof · cited by 2
- Ideal.prime_of_irreducible_absNorm_spanproof · cited by 2
- Ideal.exists_isMaximal_dvd_of_dvd_absNormproof · cited by 2
- cyclotomic_comp_X_add_one_isEisensteinAtproof · cited by 1
- cyclotomic_prime_pow_comp_X_add_one_isEisensteinAtproof · cited by 1
- Ideal.isPrime_iff_of_isPrincipalIdealRingproof · cited by 1