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Unifying some classical results on Artinian rings and modules

Donovan Leyba, Zachary Mesyan, Greg Oman

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.

Unifying some classical 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.
Paper Structure (6 sections, 11 theorems, 10 equations)

This paper contains 6 sections, 11 theorems, 10 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Artinian implies Noetherian
  4. Artinian implies countably generated
  5. Noetherian implies Artinian
  6. Cardinalities of Artinian modules

Key Result

Lemma 1

Let $R$ be a ring, let $M$ be a left $R$-module, let $N$ be a submodule of $M$, and let $\kappa$ be an infinite cardinal.

Theorems & Definitions (24)

  • Lemma 1
  • proof
  • Definition 1
  • Lemma 2
  • proof
  • Theorem 1: Hopkins-Levitzki CHJL
  • proof
  • Proposition 1
  • proof
  • Lemma 3: Sharp RS
  • ...and 14 more