Theorems · Theorem · commutative algebra
IsLocalRing.notMem_maximalIdeal
∀ {R : Type u_1} [inst : CommSemiring R] [inst_1 : IsLocalRing R] {x : R}, x ∉ IsLocalRing.maximalIdeal R ↔ IsUnit xAn element x of a commutative local semiring is not contained in the maximal ideal
iff it is a unit.
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 29 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- CommSemiringIsLocalRing
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites5
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 · cited by 4,748
- IsUnitstatement and proof · cited by 1,602
- IsLocalRingstatement and proof · cited by 339
- IsLocalRing.maximalIdealstatement · cited by 297
Cited by3
Results whose statement or proof uses this declaration.
- PowerSeries.IsWeierstrassDivisor.of_map_ne_zeroproof · cited by 11
- Valuation.isNontrivial_iff_not_a_fieldproof · cited by 1
- IsLocalRing.eq_of_eval_eq_zero_of_not_isUnit_subproof · cited by 0