# Modules, category of

The category mod- $R$ whose objects are the right unitary modules over an arbitrary associative ring $R$ with identity, and whose morphisms are the homomorphisms of $R$- 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 $R = \mathbf Z$, the ring of integers, then mod- $R$ is the category of Abelian groups, and if $R = D$ is a skew-field, then mod- $R$ is the category of vector spaces over $D$.

The properties of mod- $R$ reflect a number of important properties of the ring $R$( 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- $R$( that is, the set of natural transformations of the identity functor of the category) is isomorphic to the centre of $R$.

In ring theory, homological algebra and algebraic $K$- theory, various subcategories of the category of modules are discussed; in particular, the subcategory of finitely-generated projective $R$- modules and the associated $K$- functors (see Algebraic $K$- 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 $R$ and $S$ are Noetherian rings and if there is duality between finitely-generated right $R$- modules and finitely-generated left $S$- modules, then there is a bimodule ${} _ {S} U _ {R}$ such that the given duality is equivalent to the duality defined by the functors

$$\mathop{\rm Hom} _ {R} ( - , U ) \ \ \textrm{ and } \ \ \mathop{\rm Hom} _ {S} ( - , U ) ,$$

the ring of endomorphisms $\mathop{\rm End} U _ {R}$ is isomorphic to $S$, $\mathop{\rm End} {} _ {S} U$ is isomorphic to $R$, the bimodule $U$ is a finitely-generated injective cogenerator (both as an $R$- module and an $S$- module), and the ring $R$ 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 $R$ is quasi-Frobenius if and only if the mapping

$$M \rightarrow \mathop{\rm Hom} _ {R} ( M , R )$$

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

#### References

 [1] H. Bass, "Algebraic -theory" , Benjamin (1968) MR249491 [2] I. Bucur, A. Deleanu, "Introduction to the theory of categories and functors" , Wiley (1968) MR0236236 Zbl 0197.29205 [3] C. Faith, "Algebra: rings, modules and categories" , 1–2 , Springer (1973–1976) MR0551052 MR0491784 MR0366960 Zbl 0508.16001 Zbl 0266.16001

A duality given by a bimodule $U$ as described above is called a $U$- duality or Morita duality; cf. also (the comments to) Morita equivalence.