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 the similar way.
Contents
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 category of left modules (or that of right modules) is an abelian category. The category has 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 category of (say left) modules.
Over a commutative ring, together with the tensor product of modules ⊗, the category of modules is a symmetric monoidal category.
Example: the category of vector spaces
The category K-Vect (some authors use VectK) has all vector spaces over a fixed field K as objects and K-linear transformations 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 free vector spaces Kn, where n is any cardinal number.
Generalizations
The category of sheaves of modules over a ringed space also has enough projectives and injectives.