On rings each of whose finitely generated modules is a direct sum of cyclic modules

TL;DR

This paper characterizes non-commutative rings where every finitely generated module decomposes into cyclic modules, extending known results from the commutative case to specific classes of rings like Noetherian local and duo-rings.

Contribution

It classifies non-commutative rings with the property that all finitely generated modules are direct sums of cyclic modules, especially for Noetherian local and product rings.

Findings

Noetherian local left FGC-rings are either Artinian principal ideal rings or prime rings with principal two-sided ideals.

A Noetherian local duo-ring is a left FGC-ring if and only if it is a principal ideal ring.

Finite products of Noetherian duo-rings are left FGC-rings if and only if they are principal ideal rings.

Abstract

In this paper we study (non-commutative) rings $R$ over which every finitely generated left module is a direct sum of cyclic modules (called left FGC-rings). The commutative case was a well-known problem studied and solved in 1970s by various authors. It is shown that a Noetherian local left FGC-ring is either an Artinian principal left ideal ring, or an Artinian principal right ideal ring, or a prime ring over which every two-sided ideal is principal as a left and a right ideal. In particular, it is shown that a Noetherian local duo-ring $R$ is a left FGC-ring if and only if $R$ is a right FGC-ring, if and only if, $R$ is a principal ideal ring. Moreover, we obtain that if $R=\Pi_{i=1}^n R_i$ is a finite product of Noetherian duo-rings $R_i$ where each $R_i$ is prime or local, then $R$ is a left FGC-ring if and only if $R$ is a principal ideal ring.each $R_i$ is prime or local, then $R$ is a left FGC-ring if and only if $R$ is a principal ideal ring.