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Köthe’s classical problem posed by G. Köthe in 1935 asks to describe the rings  R  such that every left  R -module is a direct sum of cyclic modules (these rings are known as left Köthe rings). Köthe, Cohen and Kaplansky solved this problem for all commutative rings (that are Artinian principal ideal rings). During the years 1962 to 1965, Kawada solved Köthe’s problem for basic finite-dimensional algebras. But, so far, Köthe’s problem was open in the non-commutative setting. Recently, in the paper [ Several characterizations of left Köthe rings , Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. (2023) (to appear)]], we classified left Köthe rings into three classes one contained in the other:  left Köthe rings ,  strongly left Köthe rings  and  very strongly left Köthe rings , and then, we solved Köthe’s problem by giving several characterizations of these rings in terms of describing the indecomposable modules. In this paper, we will introduce the Morita duals of these notions as  left co-Köthe rings ,  strongly left co-Köthe rings  and  very strongly left co-Köthe rings , and then, we give several structural characterizations for each of them.

Left co-Köthe rings and their characterizations