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In this paper, we obtain a partial solution to the following question of Köthe: For which rings R is it true that every left (or both left and right) R-module is a direct sum of cyclic modules? Let R be a ring in which all idempotents are central. We prove that if R is a left Köthe ring (i.e., every left R-module is a direct sum of cyclic modules), then R is an Artinian principal right ideal ring. Consequently, R is a Köthe ring (i.e., each left and each right R-module is a direct sum of cyclic modules) if and only if R is an Artinian principal ideal ring. This is a generalization of a Köthe-Cohen-Kaplansky theorem.

In this paper, we define and study a valuation dimension for commutative rings. The valuation dimension is a measure of how far a commutative ring deviates from being valuation. It is shown that a ring R with valuation dimension has finite uniform dimension. We prove that a ring R is Noetherian (respectively, Artinian) if and only if the ring R × R has (respectively, finite) valuation dimension if and only if R has (respectively, finite) valuation dimension and all cyclic uniserial modules are Noetherian (respectively, Artinian). We show that the class of all rings of finite valuation dimension strictly lies between the class of Artinian rings and the class of semi-perfect rings.