# 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. 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.

**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]}

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.

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.

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**, 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.

The category of sheaves of modules over a ringed space also has enough injectives (though not always enough projectives).