Cyclic module

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over a ring , cyclic left module

A left module over isomorphic to the quotient of 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].


[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. L.A. Skornyakov (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098