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.