Unifying some classical results on Artinian rings and modules

TL;DR

This paper introduces a natural measure on modules over rings to provide concise proofs of classical theorems and new results concerning Artinian rings and modules, simplifying existing proofs and extending knowledge.

Contribution

It presents a novel measure concept on modules that streamlines proofs of classical and new theorems in Artinian ring theory.

Findings

Simplified proofs of classical theorems by Akizuki, Anderson, Hopkins, and Levitzki

Introduction of a natural measure on modules

New results on Artinian rings and modules

Abstract

In this note, we introduce a very crude but natural notion of measure on the class of left R-modules over a ring R. We use this notion to give short proofs of some classical theorems on (left) Artinian rings and modules, due to Akizuki, Anderson, Hopkins, and Levitzki, as well as of some new results.