Difference between revisions of "Cyclic module"
From Encyclopedia of Mathematics
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| − | ''over a ring | + | {{TEX|done}} |
| + | ''over a ring $R$, cyclic left module'' | ||
| − | A left module over | + | 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
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