Theorems · Theorem · commutative algebra
Module.Invertible.free_iff_linearEquiv
∀ {R : Type u} {M : Type v} [inst : CommSemiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]
[Module.Invertible R M], Module.Free R M ↔ Nonempty (M ≃ₗ[R] R)An invertible module is free iff it is isomorphic to the ring, i.e. its class is trivial in the Picard group.
- Defined in
- Mathlib.RingTheory.PicardGroup
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 98 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites30
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Modulestatement and proof · cited by 20,661
- RingHom.idstatement and proof · cited by 18,349
- AddCommMonoidstatement and proof · cited by 12,281
- CommSemiringstatement and proof · cited by 10,911
- Finsuppproof · cited by 5,255
- LinearEquivstatement and proof · cited by 3,317
- Nontrivialproof · cited by 2,416
- LinearEquiv.symmproof · cited by 1,461
- Fintype.cardproof · cited by 1,386
- Module.Freestatement and proof · cited by 597
- Module.Basis.reprproof · cited by 498
- Uniqueproof · cited by 400
Cited by3
Results whose statement or proof uses this declaration.
- Module.Invertible.finrank_eq_oneproof · cited by 2
- CommRing.Pic.mk_eq_one_iff_freeproof · cited by 2
- Module.Invertible.tensorProductComm_eq_reflproof · cited by 1