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

TL;DR

This paper investigates the structure of the reciprocal complement of polynomial rings over a field, revealing its complex algebraic properties and how they depend on the number of variables.

Contribution

It characterizes the properties of the reciprocal complement of polynomial rings in multiple variables over a field, highlighting its non-Noetherian and atomic G-domain nature.

Findings

$R(D)$ is an $n$-dimensional local G-domain.

$R(D)$ is non-Noetherian and non-factorial.

$R(D)$ has infinitely many prime ideals at each height except 0 and $n$.

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 $n\geq 2$ 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 $n$.