The category of modules is a category whose objects are right (left or two-sided, by prior arrangement) unitary modules over an arbitrary associative ring K with unity, and morphisms are homomorphisms of K-modules.
This category is the most important example of an abelian category . Moreover, for every small Abelian category, there is a complete exact embedding into a certain category of modules. The properties of the module category reflect a number of important properties of the ring , a number of important properties of the ring are associated with this category, in particular, its homological dimensions and, in part, its internal structure. The category of modules over a commutative finitely generated ring contains the entire algebraic-geometric characteristic of the affine scheme of the spectrum of the ring (one of Serre's theorems).
Categories of modules over different rings can be equivalent (that is, have the same set of classes of isomorphic objects that are in the same relation to each other). In this case, it is said that the corresponding Morita rings are equivalent . For example, the categories of modules over matrix algebras of different orders, but a common field, are equivalent to each other. All of them are equivalent to the category of spaces over the same field.
Examples
- If Is a ring of integers, then the category of modules is the category of abelian groups.
- If is a field , then the category of modules is the category of vector spaces over .
Literature
- Faith C. Algebra: Rings, Modules, Categories, Volume 1.2. - M .: Mir, 1977-79, - 688 p. + 464 p.
- Kash F. Modules and rings. - M .: "World", 1981, - 368 p.
- Lambek I. Rings and modules . - M .: "World", 1971, - 280 p.