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A Curious Characterisation of Dedekind Domains

Robert Szafarczyk

Abstract

We characterise Dedekind rings among not necessarily Noetherian domains by a property of their module homomorphisms. Our proof relies on a homological algebra argument.

A Curious Characterisation of Dedekind Domains

Abstract

We characterise Dedekind rings among not necessarily Noetherian domains by a property of their module homomorphisms. Our proof relies on a homological algebra argument.
Paper Structure (3 sections, 11 theorems, 9 equations)

This paper contains 3 sections, 11 theorems, 9 equations.

Table of Contents

  1. Introduction
  2. The characterisation
  3. Examples

Key Result

Theorem 1.1

Let $R$ be an integral domain. We say that an $R$-module homomorphism $f:M\to N$ is seemingly divisible by $r\in R$ if for every $m\in M$ we have $f(m)=rn$ for some $n\in N$ and $f(m)=0$ whenever $rm=0$. The following statements are then equivalent:

Theorems & Definitions (30)

  • Theorem 1.1
  • Proposition 1.2
  • proof
  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Theorem 2.4
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • ...and 20 more