Category of modules explained

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.

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. 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.

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

See also: compact object (a compact object in the R-mod is exactly a finitely presented module).

Category of vector spaces

See also: FinVect. The category K-Vect (some authors use 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-Vect is a special case of R-Mod (some authors use Mod_R), 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).

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
• Category of representations
• Change of rings
• Morita equivalence

References

Bibliography

• Book: Bourbaki . Algèbre . Algèbre linéaire.
• Book: Dummit . David . Foote . Richard . Abstract Algebra.
• Book: Mac Lane, Saunders . Saunders Mac Lane

. Saunders Mac Lane. Categories for the Working Mathematician . second . September 1998 . Springer . 0-387-98403-8 . 0906.18001 . 5 . .

External links

• http://ncatlab.org/nlab/show/Mod

Notes and References

1. Web site: module category in nLab. ncatlab.org.
2. trivially since any module is a quotient of a free module.