The module category 
of a ring 
is the category whose objects
are right R-modules and whose morphisms are module homomorphisms. The notation 
is used analogously for left R-modules.
A module category is an abelian category: kernels, cokernels, finite products, and finite coproducts
are formed using the corresponding module constructions.
Module Category
See also
Abelian Category, Category, Gabriel-Popescu Theorem, Module, Module Homomorphism, RingExplore with Wolfram|Alpha
References
Mac Lane, S. Categories for the Working Mathematician. New York: Springer-Verlag, 1971. https://doi.org/10.1007/978-1-4612-9839-7.Cite this as:
Weisstein, Eric W. "Module Category." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ModuleCategory.html