Theorems · Theorem · commutative algebra
IsLocalization.AtPrime.map_eq_maximalIdeal
∀ {R : Type u_1} [inst : CommSemiring R] (p : Ideal R) [inst_1 : p.IsPrime] (Rₚ : Type u_4) [inst_2 : CommSemiring Rₚ]
[inst_3 : Algebra R Rₚ] [IsLocalization.AtPrime Rₚ p] [inst_5 : IsLocalRing Rₚ],
Ideal.map (algebraMap R Rₚ) p = IsLocalRing.maximalIdeal Rₚ- Cited by
- 8 results in Mathlib
- Foundations
- Depth 86 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites15
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Algebrastatement and proof · cited by 11,388
- CommSemiringstatement and proof · cited by 10,911
- RingHomstatement · cited by 10,189
- Idealstatement and proof · cited by 4,748
- Algebra.algebraMapstatement and proof · cited by 4,706
- Ideal.IsPrimestatement and proof · cited by 827
- Ideal.mapstatement and proof · cited by 692
- Ideal.primeComplproof · cited by 462
- IsLocalRingstatement and proof · cited by 339
- IsLocalRing.maximalIdealstatement and proof · cited by 297
- Ideal.underproof · cited by 170
- IsLocalization.AtPrimestatement and proof · cited by 79
Cited by8
Results whose statement or proof uses this declaration.
- Ideal.image_subset_nonunits_valuationSubringproof · cited by 2
- IsLocalization.AtPrime.liesOver_map_of_liesOverproof · cited by 2
- LocalSubring.map_maximalIdeal_eq_top_of_isMaxproof · cited by 1
- LocalSubring.mem_of_isMax_of_isIntegralproof · cited by 1
- IsLocalization.AtPrime.radical_map_of_mem_minimalPrimesproof · cited by 1
- IsLocalization.AtPrime.ramificationIdx_map_eq_ramificationIdxproof · cited by 1
- IsLocalization.AtPrime.under_maximalIdeal_powproof · cited by 1
- Algebra.WeaklyQuasiFiniteAt.of_quasiFiniteAt_residueFieldproof · cited by 1