# Cyclic module

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].