Category of modules

In algebra, given a ring ${\displaystyle R}$, the category of left modules over ${\displaystyle R}$ is the category whose objects are all left modules over ${\displaystyle R}$ and whose morphisms are all module homomorphisms between left ${\displaystyle R}$-modules. For example, when ${\displaystyle R}$ is the ring of integers ${\displaystyle \mathbb {Z} }$, 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 ${\displaystyle R}$ but that category is equivalent to the category of left (or right) modules over the enveloping algebra of ${\displaystyle R}$ (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

The categories of left and right modules are abelian categories. These categories have enough projectives² and enough injectives.³ 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.⁴

Over a commutative ring, together with the tensor product of modules ${\displaystyle \otimes }$, the category of modules is a symmetric monoidal category.

A monoid object of the category of modules over a commutative ring ${\displaystyle R}$ is exactly an associative algebra over ${\displaystyle R}$.

A compact object in ${\displaystyle R}$-${\displaystyle \mathbf {Mod} }$ is exactly a finitely presented module.

The category ${\displaystyle K{\text{-}}\mathbf {Vect} }$ (some authors use ${\displaystyle \mathbf {Vect} _{K}}$) has all vector spaces over a field ${\displaystyle K}$ as objects, and ${\displaystyle K}$-linear maps as morphisms. Since vector spaces over ${\displaystyle K}$ (as a field) are the same thing as modules over the ring ${\displaystyle K}$, ${\displaystyle K{\text{-}}\mathbf {Vect} }$ is a special case of ${\displaystyle R}$-${\displaystyle \mathbf {Mod} }$ (some authors use ${\displaystyle \mathbf {Mod} _{R}}$), the category of left ${\displaystyle R}$-modules.

Much of linear algebra concerns the description of ${\displaystyle K{\text{-}}\mathbf {Vect} }$. For example, the dimension theorem for vector spaces says that the isomorphism classes in ${\displaystyle K{\text{-}}\mathbf {Vect} }$ correspond exactly to the cardinal numbers, and that ${\displaystyle K{\text{-}}\mathbf {Vect} }$ is equivalent to the subcategory of ${\displaystyle K{\text{-}}\mathbf {Vect} }$ which has as its objects the vector spaces ${\displaystyle K_{n}}$, where ${\displaystyle n}$ is any cardinal number.

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