E-ideals in exponential polynomial ring

TL;DR

This paper explores the structure of exponential ideals in exponential polynomial rings, revealing their independence from classical notions and the breakdown of point-ideal correspondence over algebraically closed fields.

Contribution

It introduces and characterizes exponential radical ideals and studies two notions of maximality, establishing their independence from primeness in exponential polynomial rings.

Findings

Maximal, prime, and E-maximal ideals are independent concepts.

The classical correspondence between points and maximal ideals fails over algebraically closed fields.

Introduction of exponential radical ideals and their properties.

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.