Category of modules

In algebra, given a ring "R", the category of left modules over "R" is the category whose objects are all left modules over "R" and whose morphisms are all module homomorphisms between left "R"-modules. For example, when "R" is the ring of integers Z, it is the same thing as the category of abelian groups. The category of right modules is defined in a similar way.
Note: Some authors use the term module category for the category of modules. This term can be ambiguous since it could also refer to a category with a monoidal-category action.
Properties.
The categories of left and right modules are abelian categories. These categories have enough projectives and enough injectives. Mitchell's embedding theorem states every abelian category arises as a full subcategory of the category of modules.
Projective limits and inductive limits exist in the categories of left and right modules.
Over a commutative ring, together with the tensor product of modules ⊗, the category of modules is a symmetric monoidal category.
Category of vector spaces.
The category "K"-Vect (some authors use Vect"K") has all vector spaces over a field "K" as objects, and "K"-linear maps as morphisms. Since vector spaces over "K" (as a field) are the same thing as modules over the ring "K", "K"-Vect is a special case of "R"-Mod, the category of left "R"-modules.
Much of linear algebra concerns the description of "K"-Vect. For example, the dimension theorem for vector spaces says that the isomorphism classes in "K"-Vect correspond exactly to the cardinal numbers, and that "K"-Vect is equivalent to the subcategory of "K"-Vect which has as its objects the vector spaces "K""n", where "n" is any cardinal number.
Generalizations.
The category of sheaves of modules over a ringed space also has enough injectives (though not always enough projectives).