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Difference between revisions of "Modules, category of"

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<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)</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)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> C. Faith,   "Algebra: rings, modules and categories" , '''1–2''' , Springer (1973–1976)</TD></TR></table>
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<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>
  
  

Revision as of 17:34, 31 March 2012

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) 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 as described above is called a -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=24108
This article was adapted from an original article by A.V. Mikhalev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article