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. 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.

Properties

The category of left modules (or that of right modules) is an abelian category. The category has enough projectives and enough injectives. 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.

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 Vect_K) 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

• Bourbaki, Algèbre; "Algèbre linéaire."
• Dummit, David; Foote, Richard. Abstract Algebra.
• Mac Lane, Saunders (September 1998). Categories for the Working Mathematician. Graduate Texts in Mathematics. Vol. 5 (second ed.). Springer. ISBN 0-387-98403-8. Zbl 0906.18001.

External links

• http://ncatlab.org/nlab/show/Mod