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The [[Category|category]] mod-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064480/m0644801.png" /> whose objects are the right unitary modules over an arbitrary associative ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064480/m0644802.png" /> with identity, and whose morphisms are the homomorphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064480/m0644803.png" />-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.
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If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064480/m0644804.png" />, the ring of integers, then mod-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064480/m0644805.png" /> is the category of Abelian groups, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064480/m0644806.png" /> is a skew-field, then mod-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064480/m0644807.png" /> is the category of vector spaces over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064480/m0644808.png" />.
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The properties of mod-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064480/m0644809.png" /> reflect a number of important properties of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064480/m06448010.png" /> (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-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064480/m06448011.png" /> (that is, the set of natural transformations of the identity functor of the category) is isomorphic to the centre of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064480/m06448012.png" />.
+
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.
  
In ring theory, homological algebra and algebraic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064480/m06448014.png" />-theory, various subcategories of the category of modules are discussed; in particular, the subcategory of finitely-generated projective <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064480/m06448016.png" />-modules and the associated <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064480/m06448017.png" />-functors (see [[Algebraic K-theory|Algebraic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064480/m06448018.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064480/m06448019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064480/m06448020.png" /> are Noetherian rings and if there is duality between finitely-generated right <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064480/m06448021.png" />-modules and finitely-generated left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064480/m06448022.png" />-modules, then there is a bimodule <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064480/m06448023.png" /> such that the given duality is equivalent to the duality defined by the functors
+
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 $.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064480/m06448024.png" /></td> </tr></table>
+
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 $.
  
the ring of endomorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064480/m06448025.png" /> is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064480/m06448026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064480/m06448027.png" /> is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064480/m06448028.png" />, the bimodule <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064480/m06448029.png" /> is a finitely-generated injective cogenerator (both as an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064480/m06448030.png" />-module and an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064480/m06448031.png" />-module), and the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064480/m06448032.png" /> 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]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064480/m06448033.png" /> is quasi-Frobenius if and only if the mapping
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064480/m06448034.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm Hom} _ {R} ( - , U ) \ \
 +
\textrm{ and } \ \
 +
\mathop{\rm Hom} _ {S} ( - , U ) ,
 +
$$
  
defines a duality between the categories of finitely-generated left and right <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064480/m06448035.png" />-modules.
+
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====
 
====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>
 
<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====
 
====Comments====
A duality given by a bimodule <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064480/m06448037.png" /> as described above is called a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064480/m06448039.png" />-duality or Morita duality; cf. also (the comments to) [[Morita equivalence|Morita equivalence]].
+
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]].

Latest revision as of 08:01, 6 June 2020


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

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.

How to Cite This Entry:
Modules, category of. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Modules,_category_of&oldid=47874
This article was adapted from an original article by A.V. Mikhalev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article