Theorems · Theorem · commutative algebra
Submodule.one_eq_range
∀ {R : Type u} [inst : CommSemiring R] {A : Type v} [inst_1 : Semiring A] [inst_2 : Algebra R A],
1 = (Algebra.linearMap R A).range- Defined in
- Mathlib.Algebra.Algebra.Operations
- Cited by
- 11 results in Mathlib
- Foundations
- Depth 31 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- CommSemiringSemiringAlgebra
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites11
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- RingHom.idstatement and proof · cited by 18,349
- Semiringstatement and proof · cited by 13,802
- Algebrastatement and proof · cited by 11,388
- CommSemiringstatement and proof · cited by 10,911
- LinearMapproof · cited by 10,215
- Submodulestatement and proof · cited by 7,192
- LinearMap.rangestatement and proof · cited by 893
- Algebra.linearMapstatement and proof · cited by 157
- Submodule.one_eq_spanproof · cited by 15
- LinearMap.span_singleton_eq_rangeproof · cited by 6
- LinearMap.toSpanSingleton_one_eq_algebraLinearMapproof · cited by 1
Cited by11
Results whose statement or proof uses this declaration.
- Submodule.mem_oneproof · cited by 17
- Algebra.toSubmodule_botproof · cited by 10
- Submodule.algebraMap_memproof · cited by 6
- IsLocalization.coeSubmodule_topproof · cited by 3
- Subalgebra.rank_eq_one_iffproof · cited by 3
- FractionalIdeal.den_mem_invproof · cited by 1
- FractionalIdeal.isPrincipal_of_unit_of_comap_mul_span_singleton_eq_topproof · cited by 1
- conductor_mul_differentIdealproof · cited by 1
- coeSubmodule_differentIdeal_fractionRingproof · cited by 1
- TensorAlgebra.ι_range_disjoint_oneproof · cited by 0
- Submodule.one_le_traceDual_oneproof · cited by 0