Theorems · Definition · commutative algebra
FractionalIdeal.equivNum
{R : Type u_1} →
[inst : CommRing R] →
{S : Submonoid R} →
{P : Type u_2} →
[inst_1 : CommRing P] →
[inst_2 : Algebra R P] →
[IsDomain R] →
[Module.IsTorsionFree R P] → [Nontrivial P] → {I : FractionalIdeal S P} → ↑I.den ≠ 0 → ↥↑I ≃ₗ[R] ↥I.numThe linear equivalence between the fractional ideal I and the integral ideal I.num
defined by mapping x to I.den • x.
- Defined in
- Mathlib.RingTheory.FractionalIdeal.Basic
- Cited by
- 4 results in Mathlib
- Foundations
- Depth 75 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.
- RingHom.idstatement · cited by 18,349
- CommRingstatement and proof · cited by 17,173
- Algebrastatement and proof · cited by 11,388
- Submodulestatement · cited by 7,192
- Idealstatement · cited by 4,748
- LinearEquivstatement · cited by 3,317
- Submonoidstatement and proof · cited by 3,086
- Nontrivialstatement and proof · cited by 2,416
- IsDomainstatement and proof · cited by 2,196
- Module.IsTorsionFreestatement and proof · cited by 600
- FractionalIdealstatement and proof · cited by 423
- FractionalIdeal.coeToSubmodulestatement · cited by 130
Cited by4
Results whose statement or proof uses this declaration.
- FractionalIdeal.abs_det_basis_changeproof · cited by 1
- FractionalIdeal.equivNum_applystatement · cited by 1
- FractionalIdeal.equivNum.congr_simpstatement and proof · cited by 0
- FractionalIdeal.fg_of_isNoetherianRingproof · cited by 0