Theorems · Theorem · commutative algebra
FractionalIdeal.invertible_of_principal
∀ (K : Type u_3) [inst : Field K] {R₁ : Type u_4} [inst_1 : CommRing R₁] [inst_2 : IsDomain R₁] [inst_3 : Algebra R₁ K]
[inst_4 : IsFractionRing R₁ K] (I : FractionalIdeal (nonZeroDivisors R₁) K) [(↑I).IsPrincipal], I ≠ 0 → I * I⁻¹ = 1- Cited by
- 1 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.
Cites13
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- CommRingstatement and proof · cited by 17,173
- Algebrastatement and proof · cited by 11,388
- Fieldstatement and proof · cited by 7,404
- IsDomainstatement and proof · cited by 2,196
- nonZeroDivisorsstatement and proof · cited by 895
- IsFractionRingstatement and proof · cited by 738
- FractionalIdealstatement and proof · cited by 423
- FractionalIdeal.coeToSubmodulestatement and proof · cited by 130
- Submodule.IsPrincipalstatement and proof · cited by 129
- FractionalIdeal.spanSingletonproof · cited by 73
- Submodule.IsPrincipal.generatorproof · cited by 56
- FractionalIdeal.mul_generator_self_invproof · cited by 2
Cited by1
Results whose statement or proof uses this declaration.
- FractionalIdeal.invertible_iff_generator_nonzeroproof · cited by 0