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The [[Category|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|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|Homological classification of rings]]). Connected with this category is a number of important homological invariants of the ring; in particular, its [[Homological dimension|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|Algebraic  $  K $-
theory]]). By analogy with [[Pontryagin duality|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|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|Quasi-Frobenius ring]]). A left [[Artinian ring|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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> H. Bass, "Algebraic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064480/m06448036.png" />-theory" , Benjamin (1968) {{MR|249491}} {{ZBL|}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> I. Bucur, A. Deleanu, "Introduction to the theory of categories and functors" , Wiley (1968) {{MR|0236236}} {{ZBL|0197.29205}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> C. Faith, "Algebra: rings, modules and categories" , '''1–2''' , Springer (1973–1976) {{MR|0551052}} {{MR|0491784}} {{MR|0366960}} {{ZBL|0508.16001}} {{ZBL|0266.16001}} </TD></TR></table>

====Comments====
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|Morita equivalence]].