# Cyclic module

From Encyclopedia of Mathematics

*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://www.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