Category of modules

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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 morphism are all module homomorphisms between left R-modules. The category of right modules is defined in the similar way.

Note: Some authors use the term module category for the category of modules; this term is non-standard as well as ambiguous since it could also refer to a category with a monoidal-category action.[1]


The category of left modules (or that of right modules) is an abelian category. The category has enough projectives (trivially since any module is a quotient of a free module). It also has enough injectives (showing this requires a bit of work).[2] 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.[3]

Over a commutative ring, togerher with the tensor product of modules ⊗, the category of modules is a symmetric monoidal category.

Example: the category of vector spaces[edit]

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.


The category of sheaves of modules over a ringed space also has enough projectives and injectives.

See also[edit]


  1. ^ "module category in nLab". 
  2. ^ Dummit–Foote, Ch. 10, Theorem 38.
  3. ^ Bourbaki, § 6.

External links[edit]