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.
One can also define the category of bimodules over a ring R but that category is equivalent to the category of left (or right) modules over the enveloping algebra of R (or over the opposite of that).
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 of some ring.
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.
Objects
A monoid object of the category of modules over a commutative ring R is exactly an associative algebra over R.
See also: compact object (a compact object in the R-mod is exactly a finitely presented module).
Category of vector spaces
The category K-Vect (some authors use VectK) 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 (some authors use ModR), 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 Kn, 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).
See also
Algebraic K-theory (the important invariant of the category of modules.) Category of rings Derived category Module spectrum Category of graded vector spaces Category of abelian groups Category of representations Change of rings