In this paper, we obtain a partial solution to the following question of Kothe [9]: 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 K¨othe 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 K¨othe 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 the K¨othe-Cohen-Kaplansky
theorem [3, 9].