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 the 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 category of left modules (or that of right modules) is an abelian category. The category has 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 category of (say left) modules.[4]
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.
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
References
- ↑ "module category in nLab". ncatlab.org.
- ↑ trivially since any module is a quotient of a free module.
- ↑ Dummit–Foote, Ch. 10, Theorem 38.
- ↑ Bourbaki, § 6.
- Bourbaki, Algèbre; "Algèbre linéaire."
- Dummit, David; Foote, Richard. Abstract Algebra.
- Mac Lane, Saunders (September 1998). Categories for the Working Mathematician. Graduate Texts in Mathematics. 5 (second ed.). Springer. ISBN 0-387-98403-8. Zbl 0906.18001.