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

morphism In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms a ...

s 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 In mathematics, the category Ab has the abelian groups as objects and group homomorphisms as morphisms. This is the prototype of an abelian category: indeed, every small abelian category can be embedded in Ab. Properties The zero object of Ab is ...

. 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.

Properties

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. Projective limits and inductive limits exist in the categories of left and right modules. Over a

commutative ring In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not sp ...

, together with the tensor product of modules ⊗, the category of modules is a

symmetric monoidal category In category theory, a branch of mathematics, a symmetric monoidal category is a monoidal category (i.e. a category in which a "tensor product" \otimes is defined) such that the tensor product is symmetric (i.e. A\otimes B is, in a certain strict sen ...

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, 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 In mathematics, specifically category theory, a subcategory of a category ''C'' is a category ''S'' whose objects are objects in ''C'' and whose morphisms are morphisms in ''C'' with the same identities and composition of morphisms. Intuitivel ...

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 In mathematics, a ringed space is a family of ( commutative) rings parametrized by open subsets of a topological space together with ring homomorphisms that play roles of restrictions. Precisely, it is a topological space equipped with a sheaf ...

also has enough injectives (though not always enough projectives).

References

Bibliography

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External links