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 \mathbb{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.[1]

Properties

The categories of left and right modules are abelian categories. These categories have enough projectives[2] and enough injectives.[3] Mitchell's embedding theorem states every abelian category arises as a full subcategory of the category of modules over some ring.

Projective limits and inductive limits exist in the categories of left and right modules.[4]

Over a commutative ring, together with the tensor product of modules \otimes, 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.

A compact object in R-\mathbf{Mod} is exactly a finitely presented module.

Category of vector spaces

See also: FinVect

The category K\text{-}\mathbf{Vect} (some authors use \mathbf{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\text{-}\mathbf{Vect} is a special case of R-\mathbf{Mod} (some authors use \mathbf{Mod}_R), the category of left R-modules.

Much of linear algebra concerns the description of K\text{-}\mathbf{Vect}. For example, the dimension theorem for vector spaces says that the isomorphism classes in K\text{-}\mathbf{Vect} correspond exactly to the cardinal numbers, and that K\text{-}\mathbf{Vect} is equivalent to the subcategory of K\text{-}\mathbf{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).

See also

References

  1. ^ "module category in nLab". ncatlab.org
  2. ^ trivially since any module is a quotient of a free module.
  3. ^ Dummit & Foote, pp. Ch. 10, Theorem 38.
  4. ^ Bourbaki, § 6.

Bibliography