Theorems · Theorem · commutative algebra
Submodule.span_singleton_eq_one_iff
∀ {R : Type u} [inst : CommSemiring R] {A : Type v} [inst_1 : Semiring A] [inst_2 : Algebra R A] [FaithfulSMul R A]
{x : A}, R ∙ x = 1 ↔ ∃ r, x = (algebraMap R A) ↑r- Defined in
- Mathlib.Algebra.Algebra.Operations
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 33 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites23
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement and proof · cited by 62,936
- Setstatement · cited by 53,352
- Semiringstatement and proof · cited by 13,802
- Algebrastatement and proof · cited by 11,388
- CommSemiringstatement and proof · cited by 10,911
- RingHomstatement · cited by 10,189
- Submodulestatement · cited by 7,192
- Algebra.algebraMapstatement and proof · cited by 4,706
- Unitsstatement and proof · cited by 2,804
- mul_commproof · cited by 2,262
- Units.valstatement and proof · cited by 1,966
- Submodule.spanstatement and proof · cited by 1,504
Cited by3
Results whose statement or proof uses this declaration.
- Submodule.ker_unitsToPicproof · cited by 1
- Submodule.mker_spanSingletonproof · cited by 0
- Submodule.ker_unitsMap_spanSingletonproof · cited by 0