View source for Cyclic module

{{TEX|done}}
''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 [[#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====
<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>