Category of modules explained

, the category of left modules over is the category whose objects are all left modules over and whose morphisms are all module homomorphisms between left -modules. For example, when is the ring of integers , 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

but that category is equivalent to the category of left (or right) modules over the enveloping algebra of (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.

, the category of modules is a symmetric monoidal category.

Objects

A monoid object of the category of modules over a commutative ring

is exactly an associative algebra over .

A compact object in

- is exactly a finitely presented module.

Category of vector spaces

(some authors use ) has all vector spaces over a field as objects, and

K

-linear maps as morphisms. Since vector spaces over (as a field) are the same thing as modules over the ring , is a special case of - (some authors use ), the category of left -modules.

Much of linear algebra concerns the description of

. For example, the dimension theorem for vector spaces says that the isomorphism classes in correspond exactly to the cardinal numbers, and that is equivalent to the subcategory of which has as its objects the vector spaces , where 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 representations
• Change of rings
• Morita equivalence
• Stable module category
• Eilenberg–Watts theorem

References

Bibliography

• Book: Bourbaki . Bourbaki group . 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 . .

Notes and References

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