Theorems · Theorem · commutative algebra
Submodule.LinearDisjoint.of_left_le_one_of_flat
∀ {R : Type u} {S : Type v} [inst : CommRing R] [inst_1 : Ring S] [inst_2 : Algebra R S] (M N : Submodule R S),
M ≤ 1 → ∀ [Module.Flat R ↥N], M.LinearDisjoint NIf N is flat, M is contained in i(R), where i : R → S is the structure map,
then M and N are linearly disjoint.
- Defined in
- Mathlib.LinearAlgebra.LinearDisjoint
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 103 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites8
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- CommRingstatement and proof · cited by 17,173
- Algebrastatement and proof · cited by 11,388
- Ringstatement and proof · cited by 7,463
- Submodulestatement and proof · cited by 7,192
- Module.Flatstatement and proof · cited by 279
- Submodule.LinearDisjointstatement · cited by 54
- Submodule.LinearDisjoint.of_le_left_of_flatproof · cited by 4
- Submodule.LinearDisjoint.one_leftproof · cited by 2
Cited by1
Results whose statement or proof uses this declaration.
- Ideal.eq_top_of_mk_tensor_eq_oneproof · cited by 0