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E-ideals in exponential polynomial ring

P. D'Aquino, A. Fornasiero, G. Terzo

TL;DR

The paper develops a systematic theory of exponential ideals (E-ideals) in exponential polynomial rings, introducing and contrasting three notions of maximality: primeness, maximality, and E-maximality. It shows these concepts are independent in the exponential setting and that classical correspondences (e.g., between points and maximal ideals) fail over algebraically closed fields, motivating the introduction of exponential radical ideals. By constructing and analyzing partial E-domains, free completions, and extension methods, it provides foundational results on factorization, Nullstellensatz-type phenomena, Noetherianity, and the structure of maximal E-ideals. The work also characterizes E-radical ideals via kernels, varieties, and Horn-clause logic, and highlights surprising departures from classical algebra in the exponential context, including non-Noetherianity and failure of standard Nullstellensatz results.

Abstract

We investigate exponential ideals within the context of exponential polynomial rings over exponential fields. We establish two distinct notions of maximality for exponential ideals and explore their relationship to primeness. These three concepts--prime, maximal, and E-maximal--are shown to be independent, in contrast to the classical scenario. Furthermore, we demonstrate that, over an algebraically closed field K, the correspondence between points of $K^n$ and maximal exponential ideals of the ring of exponential polynomials breaks down. Finally, we introduce and characterize exponential radical ideals. We investigate exponential ideals in the exponential polynomial ring over an exponential field. We study two notions of maximality for exponential ideals, and relate them to primeness. These three notions are independent, unlike in the classical case. We also show that over an algebraically closed field K the correspondence between points of K^n and maximal ideals of the ring of exponential polynomials does not hold.

E-ideals in exponential polynomial ring

TL;DR

The paper develops a systematic theory of exponential ideals (E-ideals) in exponential polynomial rings, introducing and contrasting three notions of maximality: primeness, maximality, and E-maximality. It shows these concepts are independent in the exponential setting and that classical correspondences (e.g., between points and maximal ideals) fail over algebraically closed fields, motivating the introduction of exponential radical ideals. By constructing and analyzing partial E-domains, free completions, and extension methods, it provides foundational results on factorization, Nullstellensatz-type phenomena, Noetherianity, and the structure of maximal E-ideals. The work also characterizes E-radical ideals via kernels, varieties, and Horn-clause logic, and highlights surprising departures from classical algebra in the exponential context, including non-Noetherianity and failure of standard Nullstellensatz results.

Abstract

We investigate exponential ideals within the context of exponential polynomial rings over exponential fields. We establish two distinct notions of maximality for exponential ideals and explore their relationship to primeness. These three concepts--prime, maximal, and E-maximal--are shown to be independent, in contrast to the classical scenario. Furthermore, we demonstrate that, over an algebraically closed field K, the correspondence between points of and maximal exponential ideals of the ring of exponential polynomials breaks down. Finally, we introduce and characterize exponential radical ideals. We investigate exponential ideals in the exponential polynomial ring over an exponential field. We study two notions of maximality for exponential ideals, and relate them to primeness. These three notions are independent, unlike in the classical case. We also show that over an algebraically closed field K the correspondence between points of K^n and maximal ideals of the ring of exponential polynomials does not hold.
Paper Structure (12 sections, 38 theorems, 77 equations)

This paper contains 12 sections, 38 theorems, 77 equations.

Table of Contents

  1. Introduction
  2. Partial E-domains and E-polynomial rings
  3. The free completion of a partial exponential ring
  4. E-ideals
  5. Maximal and Prime E-ideals
  6. Unique Factorization domain
  7. Hilbert rings and Nullstellensatz
  8. Noetherianity
  9. Strongly Maximal E-ideals Ia
  10. Construction of maximal E-ideals
  11. Exponential kernel
  12. Exponential radical ideals and characterization

Key Result

Theorem 1.1

There exists an E-maximal E-ideal of $\mathbb{C}[x]^E$ which is not prime (see Lemma maxnoprime). There exists a nonzero prime E-ideal which is not E-maximal (see Example ex:prime-not-max). There exists a prime and E-maximal E-ideal which is not strongly maximal (see Lemma lem:emax-not-max). There e

Theorems & Definitions (100)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 1.4
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Definition 2.5
  • ...and 90 more