# Category of modules

__: Category in mathematics__

**Short description**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.^{[3]} Mitchell's embedding theorem states every abelian category arises as a full subcategory of the category of modules of some ring.

Projective limits and inductive limits exist in the categories of left and right modules.^{[4]}

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

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 *K*^{n}, 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

## References

- ↑ "module category in nLab".
*ncatlab.org*. http://ncatlab.org/nlab/show/module+category. - ↑ trivially since any module is a quotient of a free module.
- ↑ Dummit & Foote, Ch. 10, Theorem 38.
- ↑ Bourbaki, § 6.

### Bibliography

- Bourbaki. "Algèbre linéaire".
*Algèbre*. - 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.

## External links

Original source: https://en.wikipedia.org/wiki/Category of modules.
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