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Wikipedia / Category of modules

In algebra, given a ring, the category of left modules is the category whose objects are all left modules and whose morphisms are all module homomorphisms is the ring of integers, 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 but that category is equivalent to the category of left (or right) modules over the enveloping algebra (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.

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 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, the category of modules is a symmetric monoidal category.

Objects

A monoid object is exactly an associative algebra.

A compact object is exactly a finitely presented module.

Category of vector spaces

The category) has all vector spaces over a field as objects, and K-linear maps (as a field) are the same thing as moduless over the ring-modules.

Much of linear algebra. For example, the dimension theorem for vector spaces says that the isomorphism classes correspond exactly to the cardinal numbers is equivalent to the subcategory is any cardinal number.

Generalizations

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

See also

References

Bibliography