Theorems · Theorem · commutative algebra
Ideal.IsPrime.mem_of_pow_mem
∀ {α : Type u} [inst : Semiring α] {I : Ideal α}, I.IsPrime → ∀ {r : α} (n : ℕ), r ^ n ∈ I → r ∈ I- Defined in
- Mathlib.RingTheory.Ideal.Prime
- Cited by
- 10 results in Mathlib
- Foundations
- Depth 29 from the axioms · uses propext, Quot.sound
- Assumes
- Semiring
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites7
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Semiringstatement and proof · cited by 13,802
- Idealstatement and proof · cited by 4,748
- pow_zeroproof · cited by 1,094
- Ideal.IsPrimestatement and proof · cited by 827
- pow_succproof · cited by 374
- Ideal.IsPrime.mem_or_memproof · cited by 22
- Ideal.IsPrime.one_notMemproof · cited by 4
Cited by10
Results whose statement or proof uses this declaration.
- PrimeSpectrum.localization_away_comap_rangeproof · cited by 17
- Ideal.IsPrime.isRadicalproof · cited by 14
- Ideal.IsPrime.pow_mem_iff_memproof · cited by 7
- LocalizedModule.subsingleton_iff_support_subsetproof · cited by 5
- Ideal.exists_disjoint_powers_of_span_eq_topproof · cited by 3
- Algebra.IsUnramifiedAt.exists_hasStandardEtaleSurjectionOnproof · cited by 1
- Ideal.isPrime_nat_iffproof · cited by 1
- Algebra.IsSmoothAt.exists_isStandardEtale_mvPolynomialproof · cited by 1
- Algebra.ZariskisMainProperty.transproof · cited by 0