Several Characterizations of Left Köthe Rings

Abstract

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 $leftK\ddot{o}thering$ 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: $leftK\ddot{o}therings$, $stronglyleftK\ddot{o}therings$ and $verystronglyleftK\ddot{o}therings$, 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.

arXiv commentThe previous version, which was long and more than 45 pages, has been organized and its defects have been fixed and it has been divided into two separate articles under the following headings. [1] Several Characterizations of Left Köthe Rings (This is the version). [2] Left Co-K{ö}the Rings and Their Characterizations (It is being prepared and will be uploaded later in the ArXiv)