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Category of modules

Category whose objects are R-modules and whose morphisms are module homomorphisms

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Source updated August 24, 2026

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 $\mathbb {Z}$ , 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 $R$ 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.

01Properties

The categories of left and right modules are abelian categories. These categories have enough projectives 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.

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

02Objects

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

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

03Category of vector spaces

The category $K{\text{-}}\mathbf {Vect}$ (some authors use $\mathbf {Vect} _{K}$ ) 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 $K$ , $K{\text{-}}\mathbf {Vect}$ is a special case of $R$ - $\mathbf {Mod}$ (some authors use $\mathbf {Mod} _{R}$ ), the category of left $R$ -modules.

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

04Generalizations

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

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Sources and credits

This article is adapted from the Wikipedia article “Category of modules”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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Category of modules

01Properties

02Objects

03Category of vector spaces

04Generalizations

Sources and credits

Related topics

Algebraic K-theory

Category of rings

Derived category