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Covering rings by proper ideals

Malcolm Hoong Wai Chen, Eric Swartz, Nicholas J. Werner

Abstract

A cover by left ideals of an associative (not necessarily commutative or unital) ring $R$ is a collection of proper left ideals whose set-theoretic union equals $R$. If such a cover exists, then $η_\ell(R)$ is the cardinality of a minimal cover, and $R$ is $η_\ell$-elementary if $η_\ell(R)<η_\ell(R/I)$ for every nonzero two-sided ideal $I$ of $R$. We classify all $η_\ell$-elementary rings, and determine their covering numbers. Covers by right or two-sided ideals are also studied. This completely characterizes rings admitting finite covers by ideals. Our results generalize to finite covers of modules by submodules, and we determine all possible covering numbers.

Covering rings by proper ideals

Abstract

A cover by left ideals of an associative (not necessarily commutative or unital) ring is a collection of proper left ideals whose set-theoretic union equals . If such a cover exists, then is the cardinality of a minimal cover, and is -elementary if for every nonzero two-sided ideal of . We classify all -elementary rings, and determine their covering numbers. Covers by right or two-sided ideals are also studied. This completely characterizes rings admitting finite covers by ideals. Our results generalize to finite covers of modules by submodules, and we determine all possible covering numbers.

Paper Structure

This paper contains 6 sections, 30 theorems, 17 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basic properties and reduction theorems
  4. An infinite family of $\eta_\ell$-elementary rings
  5. Structure of noncommutative $\eta_\ell$-elementary rings
  6. Covering numbers of modules

Key Result

Theorem 1.2

$R$ is an $\eta$-elementary ring if and only if $R^2=\{0\}$ and $(R,+) \cong C_p \times C_p$ for some prime $p$. Moreover, $\eta(R)=p+1$.

Theorems & Definitions (64)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Lemma 2.1
  • proof
  • Definition 2.2
  • Remark 2.3
  • Theorem 2.4
  • ...and 54 more