Connections between Representation-Finite and K\"othe Rings

TL;DR

This paper characterizes left k-cyclic rings and their relation to K"othe rings, extending classical theorems and providing criteria for Morita equivalence to such rings.

Contribution

It offers a new characterization of left k-cyclic rings and connects them to K"othe rings, generalizing the K"othe-Cohen-Kaplansky theorem.

Findings

Characterization of left k-cyclic rings

Criteria for rings Morita equivalent to left K"othe rings

Equivalence of Morita to artinian multiplicity-free top rings

Abstract

A ring $R$ is called left $k$-cyclic if every left $R$-module is a direct sum of indecomposable modules which are homomorphic image of $_{R}R^k$. In this paper, we give a characterization of left $k$-cyclic rings. As a consequence, we give a characterization of left K\"othe rings, which is a generalization of K\"othe-Cohen-Kaplansky theorem. We also characterize rings which are Morita equivalent to a basic left $k$-cyclic ring. As a corollary, we show that $R$ is Morita equivalent to a basic left K\"othe ring if and only if $R$ is an artinian left multiplicity-free top ring.