Shadi Asgari | Searcharxiv

arXiv2023

Left Co-Köthe Rings and Their Characterizations

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.

math.RA↗

arXiv2022

Several Characterizations of Left Köthe Rings

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 K{ö}the 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 K{ö}the rings into three categories of nested: ${\it left~Köthe~rings}$, ${\it strongly~left~K{ö}the~rings}$ and ${\it very~strongly~left~K{ö}the~rings}$, 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.

math.RA↗