Category of Modules
Get Category of Modules essential facts below. View Videos or join the Category of Modules discussion. Add Category of Modules to your PopFlock.com topic list for future reference or share this resource on social media.
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.[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.

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 ?, the category of modules is a symmetric monoidal category.

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, 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).

References

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