2022/12/28 by Shadi Asgari, Mahmood Behboodi, Asgari, Shadi +3
Chemistry · Mathematics · #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Oxidative Organic Chemistry Reactions #Representation Theory (math.RT) #Rings and Algebras (math.RA) #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.2212.13786
openalex publication_date 2022/12/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 fnite-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", submitted], 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 ring, strongly left co-Köthe rings and very strongly left co-Köthe rings, and then, we give several structural characterizations for each of them.