ModuleCat.hasKernels_moduleCat
theorem ModuleCat.hasKernels_moduleCat :
∀ {R : Type u} [inst : Ring R], CategoryTheory.Limits.HasKernels (ModuleCat R)
This theorem states that for any type `R` that forms a Ring, the category of `R`-modules has kernels. The kernels in this context are given by the inclusion of the kernel submodule into the `R`-module. In category theory, a kernel of a morphism is an important concept related to the notion of an equalizer. In the context of `R`-modules, the kernel of a module homomorphism is a submodule from which the homomorphism maps to zero.
More concisely: For any ring `R`, the category of `R`-modules has kernels given by the inclusion of the kernel submodule.
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ModuleCat.hasCokernels_moduleCat
theorem ModuleCat.hasCokernels_moduleCat :
∀ {R : Type u} [inst : Ring R], CategoryTheory.Limits.HasCokernels (ModuleCat R)
This theorem states that for any type `R` which has a ring structure, the category of `R`-modules has cokernels. The cokernel is given by the projection onto the quotient in the category of `R`-modules. In the context of category theory, the cokernel of a morphism is the analogue of the quotient by the image in linear algebra, and this theorem guarantees its existence in the category of `R`-modules for any ring `R`.
More concisely: For any ring `R`, the category of `R`-modules has cokernels, given by the quotient morphism.
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