# Modules, category of

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The category mod- whose objects are the right unitary modules over an arbitrary associative ring with identity, and whose morphisms are the homomorphisms of -modules. This category is the most important example of an Abelian category. Moreover, for every small Abelian category there is a full exact imbedding into some category of modules.

If , the ring of integers, then mod- is the category of Abelian groups, and if is a skew-field, then mod- is the category of vector spaces over .

The properties of mod- reflect a number of important properties of the ring (see Homological classification of rings). Connected with this category is a number of important homological invariants of the ring; in particular, its homological dimension. The centre of mod- (that is, the set of natural transformations of the identity functor of the category) is isomorphic to the centre of .

In ring theory, homological algebra and algebraic -theory, various subcategories of the category of modules are discussed; in particular, the subcategory of finitely-generated projective -modules and the associated -functors (see Algebraic -theory). By analogy with Pontryagin duality, dualities between full subcategories of the category of modules have been studied; in particular between subcategories of finitely-generated modules. For example, it has been established that if and are Noetherian rings and if there is duality between finitely-generated right -modules and finitely-generated left -modules, then there is a bimodule such that the given duality is equivalent to the duality defined by the functors

the ring of endomorphisms is isomorphic to , is isomorphic to , the bimodule is a finitely-generated injective cogenerator (both as an -module and an -module), and the ring is semi-perfect (cf. Semi-perfect ring). The most important class of rings, arising in the consideration of duality of modules, is the class of quasi-Frobenius rings (cf. Quasi-Frobenius ring). A left Artinian ring is quasi-Frobenius if and only if the mapping

defines a duality between the categories of finitely-generated left and right -modules.

#### References

 [1] H. Bass, "Algebraic -theory" , Benjamin (1968) [2] I. Bucur, A. Deleanu, "Introduction to the theory of categories and functors" , Wiley (1968) [3] C. Faith, "Algebra: rings, modules and categories" , 1–2 , Springer (1973–1976)