Theorems · Theorem · commutative algebra
Ideal.eq_top_of_isUnit_mem
∀ {α : Type u} [inst : Semiring α] (I : Ideal α) {x : α}, x ∈ I → IsUnit x → I = ⊤- Defined in
- Mathlib.RingTheory.Ideal.Lattice
- Cited by
- 26 results in Mathlib
- Foundations
- Depth 26 from the axioms · uses propext, Quot.sound
- Assumes
- Semiring
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.
- Semiringstatement and proof · cited by 13,802
- Top.topstatement and proof · cited by 9,680
- Idealstatement and proof · cited by 4,748
- IsUnitstatement and proof · cited by 1,602
- IsUnit.exists_left_invproof · cited by 10
- Ideal.eq_top_of_unit_memproof · cited by 2
Cited by26
Results whose statement or proof uses this declaration.
- Ring.krullDimLE_zero_and_isLocalRing_tfaeproof · cited by 7
- IsLocalization.disjoint_under_iffproof · cited by 6
- IsLocalization.isPrime_iff_isPrime_disjointproof · cited by 6
- IsLocalRing.of_unique_max_idealproof · cited by 4
- IsLocalization.AtPrime.isUnit_to_map_iffproof · cited by 4
- Submodule.IsPrincipal.prime_generator_of_isPrimeproof · cited by 4
- Valuation.Uniformizer.is_generatorproof · cited by 4
- IsLocalization.isLocalization_isLocalization_atPrime_isLocalizationproof · cited by 3
- IsLocalization.isMaximal_iff_isMaximal_disjointproof · cited by 3
- Ideal.notMem_of_isUnitproof · cited by 3
- IsLocalRing.isMaximal_iffproof · cited by 3