Theorems · Theorem · commutative algebra
Submodule.LinearDisjoint.of_le_left_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.LinearDisjoint N → ∀ {M' : Submodule R S}, M' ≤ M → ∀ [Module.Flat R ↥N], M'.LinearDisjoint NIf M and N are linearly disjoint, if N is flat, then for any submodule M' of M,
M' and N are also linearly disjoint.
- Defined in
- Mathlib.LinearAlgebra.LinearDisjoint
- Cited by
- 4 results in Mathlib
- Foundations
- Depth 102 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites20
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
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- RingHom.idproof · cited by 18,349
- CommRingstatement and proof · cited by 17,173
- Algebrastatement and proof · cited by 11,388
- LinearMapproof · cited by 10,215
- Ringstatement and proof · cited by 7,463
- Submodulestatement and proof · cited by 7,192
- TensorProductproof · cited by 2,545
- LinearMap.compproof · cited by 1,642
- LinearMap.extproof · cited by 844
- Module.Flatstatement and proof · cited by 279
- LinearMap.rTensorproof · cited by 266
Cited by4
Results whose statement or proof uses this declaration.
- Subalgebra.LinearDisjoint.of_le_left_of_flatproof · cited by 3
- Submodule.LinearDisjoint.of_left_le_one_of_flatproof · cited by 1
- Submodule.LinearDisjoint.of_le_of_flat_leftproof · cited by 0
- Submodule.LinearDisjoint.of_le_of_flat_rightproof · cited by 0