The reciprocal complements of classes of integral domains

TL;DR

This paper investigates the properties of reciprocal complements of integral domains, focusing on prime ideals, localizations, Krull dimension, and specific classes like semigroup algebras, providing new characterizations and structural insights.

Contribution

It introduces a detailed study of reciprocal complements of integral domains, including their prime ideal structure, localizations, and conditions under which they form DVRs, expanding understanding of these ring constructions.

Findings

Characterization of when R(D) is a DVR

Descriptions of reciprocal complements for semigroup algebras

Analysis of prime ideals and Krull dimension in R(D)

Abstract

Given an integral domain $D$ with quotient field $\mathcal{Q}(D)$, the reciprocal complement of $D$ is the subring $R(D)$ of $\mathcal{Q}(D)$ whose elements are all the sums $\frac{1}{d_1}+\ldots+\frac{1}{d_n} $ for $d_1, \ldots, d_n$ nonzero elements of $D$. In this article we study problems related with prime ideals, localizations and Krull dimension of rings of the form $R(D)$ and we describe the reciprocal complements of classes of domains, including semigroup algebras and $ D+ \mathfrak{m} $ constructions. We also characterize when $R(D)$ is a DVR.