Theorems · Theorem · commutative algebra
Submodule.LinearDisjoint.map
∀ {R : Type u} {S : Type v} [inst : CommSemiring R] [inst_1 : Semiring S] [inst_2 : Algebra R S] {M N : Submodule R S},
M.LinearDisjoint N →
∀ {T : Type w} [inst_3 : Semiring T] [inst_4 : Algebra R T] (f : S →ₐ[R] T),
Function.Injective ⇑f → (Submodule.map (↑f) M).LinearDisjoint (Submodule.map (↑f) N)Linear disjointness is preserved by injective algebra homomorphisms.
- Defined in
- Mathlib.LinearAlgebra.LinearDisjoint
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 73 from the axioms · uses propext, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites17
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
- RingHom.idstatement · 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
- Submodulestatement and proof · cited by 7,192
- AlgHomstatement and proof · cited by 3,236
- TensorProductproof · cited by 2,545
- Submodule.mapstatement and proof · cited by 614
- SemilinearMapClass.semilinearMapstatement and proof · cited by 80
- Submodule.LinearDisjointstatement and proof · cited by 54
- Submodule.mulMapproof · cited by 43
Cited by1
Results whose statement or proof uses this declaration.
- Subalgebra.LinearDisjoint.mapproof · cited by 4