PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 142, Number 8, August 2014, Pages 2625–2631 S 0002-9939(2014)11158-0 Article electronically published on April 22, 2014 ON LEFT K ¨ OTHE RINGS AND A GENERALIZATION OF AK ¨ OTHE-COHEN-KAPLANSKY THEOREM M. BEHBOODI, A. GHORBANI, A. MORADZADEH-DEHKORDI, AND S. H. SHOJAEE (Communicated by Birge Huisgen-Zimmermann) Abstract. In this paper, we obtain a partial solution to the following question of K¨othe: 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 a K¨othe-Cohen-Kaplansky theorem. 1. Introduction All rings have identity elements and all modules are unital. A ring R is local in case R has a unique left maximal ideal. An Artinian (resp. Noetherian) ring is a ring which is both a left and right Artinian (resp. Noetherian). A principal ideal ring is a ring which is both a left and a right principal ideal ring. Also, a ring whose lattice of left ideals is linearly ordered under inclusion is called a left uniserial ring. A uniserial ring is a ring which is both left and right uniserial. Note that left and right uniserial rings are in particular local rings, and commutative uniserial rings are also known as valuation rings. In [9] K¨othe proved the following result. Result 1.1 (K¨ othe, [9]). Over an Artinian principal ideal ring, each module is a direct sum of cyclic modules. Furthermore, if a commutative Artinian ring has the property that all its modules are direct sums of cyclic modules, then it is necessarily a principal ideal ring. Later Cohen and Kaplansky [3] obtained the following result. Result 1.2 (Cohen and Kaplansky, [3]). If R is a commutative ring such that each R-module is a direct sum of cyclic modules, then R must be an Artinian principal ideal ring. Received by the editors June 15, 2010 and, in revised form, March 6, 2011; March 28, 2011; and August 27, 2012. 2010 Mathematics Subject Classification. Primary 16D10, 16D70, 16P20; Secondary 16N60. Key words and phrases. Cyclic modules K¨ othe rings, principal ideal rings, uniserial rings. The research of the first author was in part supported by a grant from IPM (No. 89160031). The first author is the corresponding author. c 2014 American Mathematical Society Reverts to public domain 28 years from publication 2625 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use