Theorems · Theorem · commutative algebra
IsLocalRing.of_unique_max_ideal
∀ {R : Type u_1} [inst : CommSemiring R], (∃! I, I.IsMaximal) → IsLocalRing RA semiring is local if it has a unique maximal ideal.
- Defined in
- Mathlib.RingTheory.LocalRing.Basic
- Cited by
- 4 results in Mathlib
- Foundations
- Depth 76 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.
Cites12
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
- IsUnitproof · cited by 1,602
- Ideal.IsMaximalstatement and proof · cited by 452
- IsLocalRingstatement · cited by 339
- ExistsUniquestatement and proof · cited by 268
- Ideal.add_memproof · cited by 36
- nonunitsproof · cited by 35
- Ideal.eq_top_of_isUnit_memproof · cited by 26
- Ideal.IsMaximal.outproof · cited by 16
- exists_max_ideal_of_mem_nonunitsproof · cited by 3
- IsLocalRing.of_nonunits_addproof · cited by 2
Cited by4
Results whose statement or proof uses this declaration.
- Ring.krullDimLE_zero_and_isLocalRing_tfaeproof · cited by 7
- IsLocalRing.of_singleton_maximalSpectrumproof · cited by 2
- isLocalRing_of_isAdicComplete_maximalproof · cited by 1
- IsLocalRing.of_unique_nonzero_primeproof · cited by 1