mathlib3 documentation

algebra.category.Module.epi_mono

Monomorphisms in Module R #

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This file shows that an R-linear map is a monomorphism in the category of R-modules if and only if it is injective, and similarly an epimorphism if and only if it is surjective.

source
theorem Module.ker_eq_bot_of_mono {R : Type u} [ring R] {X Y : Module R} (f : X ⟶ Y) [category_theory.mono f] :
linear_map.ker f = ⊥
source
theorem Module.range_eq_top_of_epi {R : Type u} [ring R] {X Y : Module R} (f : X ⟶ Y) [category_theory.epi f] :
linear_map.range f = ⊤
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theorem Module.mono_iff_ker_eq_bot {R : Type u} [ring R] {X Y : Module R} (f : X ⟶ Y) :
category_theory.mono f ↔ linear_map.ker f = ⊥
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theorem Module.mono_iff_injective {R : Type u} [ring R] {X Y : Module R} (f : X ⟶ Y) :
category_theory.mono f ↔ function.injective ⇑f
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theorem Module.epi_iff_range_eq_top {R : Type u} [ring R] {X Y : Module R} (f : X ⟶ Y) :
category_theory.epi f ↔ linear_map.range f = ⊤
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theorem Module.epi_iff_surjective {R : Type u} [ring R] {X Y : Module R} (f : X ⟶ Y) :
category_theory.epi f ↔ function.surjective ⇑f
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def Module.unique_of_epi_zero {R : Type u} [ring R] {M : Type v} [add_comm_group M] [module R M] (X : Module R) [h : category_theory.epi 0] :
unique M

If the zero morphism is an epi then the codomain is trivial.

Equations
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@[protected, instance]
def Module.mono_as_hom'_subtype {R : Type u} [ring R] {X : Module R} (U : submodule R ↥X) :
category_theory.mono (Module.as_hom_right U.subtype)
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@[protected, instance]
def Module.epi_as_hom''_mkq {R : Type u} [ring R] {X : Module R} (U : submodule R ↥X) :
category_theory.epi (Module.as_hom_left U.mkq)
source
@[protected, instance]
def Module.forget_preserves_epimorphisms {R : Type u} [ring R] :
(category_theory.forget (Module R)).preserves_epimorphisms
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@[protected, instance]
def Module.forget_preserves_monomorphisms {R : Type u} [ring R] :
(category_theory.forget (Module R)).preserves_monomorphisms