An introduction to reciprocal complements of integral domains

TL;DR

This paper surveys the concept of reciprocal complements in integral domains, providing an overview of current research and presenting new findings and connections in the field.

Contribution

It offers a comprehensive survey of reciprocal complements and introduces new results and connections related to their structure in integral domains.

Findings

Updated classification of reciprocal complements

New connections between reciprocal complements and other algebraic structures

Extended results on the properties of reciprocal complements

Abstract

Given an integral domain $D$ with fraction field $F$, its *reciprocal complement* is the subring of $F$ generated by all $1/d$ for nonzero $d$ in $D$. This paper serves doubly as a survey of the current state of the field and an update with new results and connections.