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The reciprocal complements of classes of integral domains

Lorenzo Guerrieri

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.

The reciprocal complements of classes of integral domains

Abstract

Given an integral domain with quotient field , the reciprocal complement of is the subring of whose elements are all the sums for nonzero elements of . In this article we study problems related with prime ideals, localizations and Krull dimension of rings of the form and we describe the reciprocal complements of classes of domains, including semigroup algebras and constructions. We also characterize when is a DVR.

Paper Structure

This paper contains 6 sections, 27 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. Prime ideals of reciprocal complements
  3. Finitely generated $K$-algebras and Krull dimension of reciprocal complements
  4. Semigroup algebras
  5. Reciprocal complements of $D+\mathfrak{m}$ constructions
  6. The case when $R(D)$ is a DVR

Key Result

Theorem 2.1

EGL Let $D$ be an integral domain. Then, the ring $R(D)$ is local and its maximal ideal is generated by all the elements $\frac{1}{d}$ where $d \in D$ is not an Egyptian element.

Theorems & Definitions (64)

  • Conjecture 1.1
  • Theorem 2.1
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • Corollary 2.4
  • proof
  • Lemma 2.5
  • proof
  • Corollary 2.6
  • ...and 54 more