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 morphism are all module homomorphisms between left R-modules. The category of right modules is defined in the similar way.
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). Mitchell's embedding theorem states every abelian category arises as a full subcategory of the category of modules.
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.
- 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
- 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.
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