Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Ring theory in o-minimal structures

Annalisa Conversano

TL;DR

We address the problem of describing rings definable in any o-minimal structure ${\mathcal M}$. Our approach develops a general ring theory in the o-minimal setting and shows that definable rings are assembled from finite-dimensional associative ${\mathbb R}$-algebras and finite rings annihilated on the connected component, with definable unitization providing a smallest unital closure. In particular, every definably connected ring with nontrivial multiplication defines an infinite field and simple definable rings are matrix rings ${M_n(D)}$ over definable division rings $D$ that are definable over a real closed field; semiprime rings decompose into a finite product of such simples. The unital case yields that all ideals are definable, making unital rings Artinian and Noetherian, while the general case is governed by a Jacobson-radical decomposition ${R=J(R)\oplus S}$ with ${J(R)}$ nilpotent and ${S}$ semiprime, plus reductions to connected components and finite parts. Together, these results provide a complete, constructive classification of definable rings in o-minimal structures, linking them to classical finite-dimensional algebras and matrix rings over definable division algebras.

Abstract

We develop a general ring theory in the o-minimal setting culminating in a description of all the definable rings in an arbitrary o-minimal structure. We show that every definably connected ring with non-trivial multiplication defines an infinite field and it is essentially semialgebraic. A surprisingly strong correspondence between definably connected rings and finite-dimensional associative $\mathbb{R}$-algebras is established. Every ideal of a definable unital ring is definable, from which it follows that every definable unital ring is Artinian and Noetherian. If a definable ring $R$ is not unital, we give necessary and sufficient conditions for $R$ to embed in a definable unital ring as an ideal. Moreover, when this is the case, we provide the smallest such definable unital ring $R^{\wedge}$, its definable unitazation.

Ring theory in o-minimal structures

TL;DR

We address the problem of describing rings definable in any o-minimal structure . Our approach develops a general ring theory in the o-minimal setting and shows that definable rings are assembled from finite-dimensional associative -algebras and finite rings annihilated on the connected component, with definable unitization providing a smallest unital closure. In particular, every definably connected ring with nontrivial multiplication defines an infinite field and simple definable rings are matrix rings over definable division rings that are definable over a real closed field; semiprime rings decompose into a finite product of such simples. The unital case yields that all ideals are definable, making unital rings Artinian and Noetherian, while the general case is governed by a Jacobson-radical decomposition with nilpotent and semiprime, plus reductions to connected components and finite parts. Together, these results provide a complete, constructive classification of definable rings in o-minimal structures, linking them to classical finite-dimensional algebras and matrix rings over definable division algebras.

Abstract

We develop a general ring theory in the o-minimal setting culminating in a description of all the definable rings in an arbitrary o-minimal structure. We show that every definably connected ring with non-trivial multiplication defines an infinite field and it is essentially semialgebraic. A surprisingly strong correspondence between definably connected rings and finite-dimensional associative -algebras is established. Every ideal of a definable unital ring is definable, from which it follows that every definable unital ring is Artinian and Noetherian. If a definable ring is not unital, we give necessary and sufficient conditions for to embed in a definable unital ring as an ideal. Moreover, when this is the case, we provide the smallest such definable unital ring , its definable unitazation.

Paper Structure

This paper contains 7 sections, 33 theorems, 123 equations.

Table of Contents

  1. Introduction
  2. Some general results
  3. Nilpotent rings
  4. Reduced rings
  5. The Jacobson radical
  6. Semiprime rings
  7. The general case

Key Result

Theorem 1

Let $\mathcal{M}$ be an o-minimal expansion of a field. The $\mathcal{M}$-definable rings are:

Theorems & Definitions (75)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Proposition 7
  • Corollary 8
  • Corollary 9
  • Corollary 2.2
  • ...and 65 more