Left Co-K\"othe Rings and Their Characterizations

TL;DR

This paper introduces and characterizes left co-K"othe rings, the Morita duals of left K"othe rings, providing structural insights and classifications in the context of non-commutative ring theory.

Contribution

It defines Morita duals of left K"othe rings and offers structural characterizations, advancing the understanding of module decompositions in non-commutative rings.

Findings

Defined left co-K"othe, strongly left co-K"othe, very strongly left co-K"othe rings.

Provided structural characterizations for each class.

Extended K"othe's problem to non-commutative rings via Morita duals.

Abstract

K\"othe's classical problem posed by G. K\"othe 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 K\"othe rings). K\"othe, Cohen and Kaplansky solved this problem for all commutative rings (that are Artinian principal ideal rings). During the years 1962 to 1965, Kawada solved K\"othe's problem for basic fnite-dimensional algebras. But, so far, K\"othe's problem was open in the non-commutative setting. Recently, in the paper ["Several characterizations of left K\"othe rings", submitted], we classified left K\"othe rings into three classes one contained in the other: left K\"othe rings, strongly left K\"othe rings and very strongly left K\"othe rings, and then, we solved K\"othe'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-K\"othe ring, strongly left co-K\"othe rings and very strongly left co-K\"othe rings, and then, we give several structural characterizations for each of them.