Cyclic-Uniform Uniserial Modules and Rings

TL;DR

This paper introduces and studies cyclic-uniform uniserial modules and rings, generalizing virtually uniserial modules by replacing the virtually simple condition with a cyclic uniform condition, and characterizes certain Noetherian rings with this property.

Contribution

It generalizes virtually uniserial modules by defining cyclic-uniform uniserial modules and characterizes Noetherian rings where all finitely generated modules are cyclic-uniform serial.

Findings

Characterization of Noetherian left cyclic-uniform uniserial rings

Properties of cyclic-uniform (uni)serial modules and rings

Identification of rings where all finitely generated modules are cyclic-uniform serial

Abstract

An $R$-module $M$ is called virtually uniserial if for every finitely generated submodule $0 \neq K \subseteq M$, $K/$Rad$(K)$ is virtually simple. In this paper, we generalize virtually uniserial modules by dropping the virtually simple condition and replacing it by the cyclic uniform condition. An $R$-module $M$ is called cyclic-uniform uniserial if $K/$Rad$(K)$ is cyclic and uniform, for every finitely generated submodule $0 \neq K \subseteq M$. Also, $M$ is said to be cyclic-uniform serial if it is a direct sum of cyclic-uniform uniserial modules. Several properties of cyclic-uniform (uni)serial modules and rings are given. Moreover, the structure of Noetherian left cyclic-uniform uniserial rings are characterized. Finally, we study rings $R$ have the property that every finitely generated $R$-module is cyclic-uniform serial.