2022/06/13 by Asgari, Shadi, Behboodi, Mahmood, Khedrizadeh, Somayeh
Mathematics · #16D10 #16D70 #16D90 (Primary) #16G60 #16P20 (Secondary) #Algebraic structures and combinatorial models #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Representation Theory (math.RT) #Rings and Algebras (math.RA) #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.2206.06453
openalex publication_date 2022/06/13 · openalex created_date 2022/07/21 · openalex updated_date 2026/07/28
We study the classical Köthe's problem, concerning the structure of non-commutative rings with the property that: ``every left module is a direct sum of cyclic modules". In 1934, Köthe showed that left modules over Artinian principal ideal rings are direct sums of cyclic modules. A ring R is called a \it left~Köthe~ring if every left R-module is a direct sum of cyclic R-modules. In 1951, Cohen and Kaplansky proved that all commutative Köthe rings are Artinian principal ideal rings. During the years 1962 to 1965, Kawada solved the Köthe's problem for basic fnite-dimensional algebras: Kawada's theorem characterizes completely those finite-dimensional algebras for which any indecomposable module has square-free socle and square-free top, and describes the possible indecomposable modules. But, so far, the Köthe's problem is open in the non-commutative setting. In this paper, we break the class of left Köthe rings into three categories of nested: \it left~Köthe~rings, \it strongly~left~Köthe~rings and \it very~strongly~left~Köthe~rings, and then, we solve the Köthe's problem by giving several characterizations of these rings in terms of describing the indecomposable modules. Finally, we give a new generalization of Köthe-Cohen-Kaplansky theorem.