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Note on radical and prime E-ideals

Antongiulio Fornasiero, Giuseppina Terzo

TL;DR

The paper investigates E-ideals in E-rings, focusing on exponential polynomials and their Noetherian behavior. It proves that the E-polynomial ring $\mathbb{C}[\bar{x}]^E$ is not Noetherian even when restricted to prime E-ideals, using an explicit construction of a chain of prime E-ideals. It develops the notion of E-radical ideals, establishing that $\mathrm{E\text{-}rad}(J)$ coincides with a syntactic radical $\mathrm{T_2\text{-}rad}(J)$ and with the algebraic radical $\sqrt[\!E\]{J}$ via an explicit, recursive description involving $\sqrt[n]{X}$ and $\sqrt[E]{X}$. The work also provides a framework to extend and lift primeness through the E-polynomial construction and situates E-radical ideals within a quasi-variety of E-reduced rings, yielding constructive axiomatizations. Overall, these results deepen the understanding of exponential-structure ideals and their geometric interpretation, with implications for the model theory of exponential rings and related decision problems.

Abstract

We show that the ring of exponential polynomials is not Noetherian even respect to prime E-ideals. Moreover we give a characterization of exponential radical ideals

Note on radical and prime E-ideals

TL;DR

The paper investigates E-ideals in E-rings, focusing on exponential polynomials and their Noetherian behavior. It proves that the E-polynomial ring is not Noetherian even when restricted to prime E-ideals, using an explicit construction of a chain of prime E-ideals. It develops the notion of E-radical ideals, establishing that coincides with a syntactic radical and with the algebraic radical via an explicit, recursive description involving and . The work also provides a framework to extend and lift primeness through the E-polynomial construction and situates E-radical ideals within a quasi-variety of E-reduced rings, yielding constructive axiomatizations. Overall, these results deepen the understanding of exponential-structure ideals and their geometric interpretation, with implications for the model theory of exponential rings and related decision problems.

Abstract

We show that the ring of exponential polynomials is not Noetherian even respect to prime E-ideals. Moreover we give a characterization of exponential radical ideals
Paper Structure (7 sections, 13 theorems, 17 equations)

This paper contains 7 sections, 13 theorems, 17 equations.

Table of Contents

  1. Introduction
  2. E-polynomial ring and basic results
  3. Partial E-rings and E-ideals
  4. Construction of E-polynomial ring
  5. Noetherianity
  6. Exponential radical ideals
  7. Characterization of radical E-ideals

Key Result

Lemma 2.5

Let $R$ be an E-ring and $S$ be a partial subring of $R[\bar{x}]^E,$ and assume that $S = R[\bar{x}][e^A]$ for some $\mathbb{Q}$-linear subspace $A$ of $R[\bar{x}]$ which has trivial intersection with $R$. Then, $S^E = R[\bar{x}]^E$.

Theorems & Definitions (37)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Lemma 2.5: ideali*Lemma 2.10
  • Lemma 2.6
  • Definition 3.1
  • Theorem 3.2
  • Lemma 3.3
  • proof
  • ...and 27 more