The reciprocal complement of a polynomial ring in several variables over a field

Neil Epstein, Lorenzo Guerrieri and K. Alan Loper

Vol. 338 (2025), No. 2, 267–293

DOI: 10.2140/pjm.2025.338.267

Abstract

The reciprocal complement $R(D)$ of an integral domain $D$ is the subring of its fraction field generated by the reciprocals of its nonzero elements. Many properties of $R(D)$ are determined when $D$ is a polynomial ring in $n\ge 2$ variables over a field. In particular, $R(D)$ is an $n$-dimensional, local, non-Noetherian, non-integrally closed, non-factorial, atomic G-domain, with infinitely many prime ideals at each height other than $0$ and $n$.

Keywords

reciprocal complement, Egyptian fraction, polynomial rings

Mathematical Subject Classification

Primary: 13F99

Secondary: 13A18, 13F20, 13G05

Milestones

Received: 31 July 2024

Revised: 18 April 2025

Accepted: 24 June 2025

Published: 24 August 2025