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Difference between revisions of "Cyclic module"

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''over a ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027520/c0275201.png" />, cyclic left module''
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''over a ring $R$, cyclic left module''
  
A left module over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027520/c0275202.png" /> isomorphic to the quotient of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027520/c0275203.png" /> by some left ideal. In particular, irreducible modules are cyclic. The Köthe problem is connected with cyclic modules (see [[#References|[4]]]): Over what rings is every (or every finitely generated) module isomorphic to a direct sum of cyclic modules? In the class of commutative rings these turn out to be exactly the Artinian principal ideal rings (see [[#References|[1]]], [[#References|[3]]]). There is also a complete description of commutative rings over which every finitely generated module splits into a direct sum of cyclic modules [[#References|[2]]].
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A left module over $R$ isomorphic to the quotient of $R$ by some left ideal. In particular, irreducible modules are cyclic. The Köthe problem is connected with cyclic modules (see [[#References|[4]]]): Over what rings is every (or every finitely generated) module isomorphic to a direct sum of cyclic modules? In the class of commutative rings these turn out to be exactly the Artinian principal ideal rings (see [[#References|[1]]], [[#References|[3]]]). There is also a complete description of commutative rings over which every finitely generated module splits into a direct sum of cyclic modules [[#References|[2]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  C. Faith,  "Algebra: rings, modules and categories" , '''1–2''' , Springer  (1973–1977)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  W. Brandal,  "Commutative rings whose finitely generated modules decompose" , Springer  (1979)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  C. Faith,  "The basis theorem for modules: a brief survey and a look to the future" , ''Ring theory, Proc. Antwerp Conf. 1977'' , M. Dekker  (1978)  pp. 9–23</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  G. Köthe,  "Verallgemeinerte Abelsche Gruppen mit hyperkomplexem Operatorenring"  ''Math. Z.'' , '''39'''  (1935)  pp. 31–44</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  C. Faith,  "Algebra: rings, modules and categories" , '''1–2''' , Springer  (1973–1977)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  W. Brandal,  "Commutative rings whose finitely generated modules decompose" , Springer  (1979)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  C. Faith,  "The basis theorem for modules: a brief survey and a look to the future" , ''Ring theory, Proc. Antwerp Conf. 1977'' , M. Dekker  (1978)  pp. 9–23</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  G. Köthe,  "Verallgemeinerte Abelsche Gruppen mit hyperkomplexem Operatorenring"  ''Math. Z.'' , '''39'''  (1935)  pp. 31–44</TD></TR></table>

Latest revision as of 17:29, 30 April 2014

over a ring $R$, cyclic left module

A left module over $R$ isomorphic to the quotient of $R$ by some left ideal. In particular, irreducible modules are cyclic. The Köthe problem is connected with cyclic modules (see [4]): Over what rings is every (or every finitely generated) module isomorphic to a direct sum of cyclic modules? In the class of commutative rings these turn out to be exactly the Artinian principal ideal rings (see [1], [3]). There is also a complete description of commutative rings over which every finitely generated module splits into a direct sum of cyclic modules [2].

References

[1] C. Faith, "Algebra: rings, modules and categories" , 1–2 , Springer (1973–1977)
[2] W. Brandal, "Commutative rings whose finitely generated modules decompose" , Springer (1979)
[3] C. Faith, "The basis theorem for modules: a brief survey and a look to the future" , Ring theory, Proc. Antwerp Conf. 1977 , M. Dekker (1978) pp. 9–23
[4] G. Köthe, "Verallgemeinerte Abelsche Gruppen mit hyperkomplexem Operatorenring" Math. Z. , 39 (1935) pp. 31–44
How to Cite This Entry:
Cyclic module. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cyclic_module&oldid=32001
This article was adapted from an original article by L.A. Skornyakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article