Theorems · Theorem · commutative algebra
Submodule.linearDisjoint_iff
∀ {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 ↔ Function.Injective ⇑(M.mulMap N)- Defined in
- Mathlib.LinearAlgebra.LinearDisjoint
- Cited by
- 10 results in Mathlib
- Foundations
- Depth 61 from the axioms · uses propext, 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.
- 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
- LinearMapstatement · cited by 10,215
- Submodulestatement and proof · cited by 7,192
- TensorProductstatement · cited by 2,545
- Submodule.LinearDisjointstatement and proof · cited by 54
- Submodule.mulMapstatement and proof · cited by 43
- Submodule.LinearDisjoint.casesOnproof · cited by 1
Cited by10
Results whose statement or proof uses this declaration.
- Submodule.LinearDisjoint.symm_of_commuteproof · cited by 3
- Submodule.LinearDisjoint.one_leftproof · cited by 2
- Submodule.LinearDisjoint.one_rightproof · cited by 2
- Subalgebra.LinearDisjoint.of_finrank_sup_of_freeproof · cited by 2
- Subalgebra.LinearDisjoint.include_rangeproof · cited by 2
- Subalgebra.LinearDisjoint.of_linearDisjoint_finite_leftproof · cited by 2
- Submodule.LinearDisjoint.of_linearDisjoint_fg_leftproof · cited by 1
- Submodule.LinearDisjoint.of_linearDisjoint_fg_rightproof · cited by 1
- Subalgebra.linearDisjoint_iff_injectiveproof · cited by 1
- Submodule.LinearDisjoint.mapproof · cited by 1