What are discrete valuation rings? What are Dedekind domains?

TL;DR

This paper provides an extended introduction to discrete valuation rings and Dedekind domains, discussing their properties and generalizations like almost Dedekind, Prüfer, and Krull domains, with brief insights into singular integral domains of dimension 1.

Contribution

It offers a comprehensive overview of discrete valuation rings and Dedekind domains, including recent generalizations and related domain classes.

Findings

Clarifies the definitions and properties of Dedekind domains and discrete valuation rings.

Introduces generalizations such as almost Dedekind, Prüfer, and Krull domains.

Provides brief discussion on singular integral domains of dimension 1.

Abstract

This is an extended introduction to discrete valuation rings and Dedekind domains. Some natural generalizations of Dedekind domains are also (briefly) discussed including "almost Dedekind domains", Pr\"ufer domains, Krull domains, and singular integral domains of dimension 1.