On the spectrum of rings of functions

TL;DR

This paper investigates the structure of maximal ideals in rings of functions, especially integer-valued polynomials, by examining their relation to ultrapowers of residue class rings, providing insights into their algebraic properties.

Contribution

It characterizes which maximal ideals in rings of functions originate from ultrapowers of residue class rings, extending understanding of their algebraic structure.

Findings

Identifies conditions for maximal ideals to come from ultrapowers

Provides a framework for analyzing subrings of product rings

Enhances understanding of the spectrum of rings of functions

Abstract

Let $D$ be a domain and $M$ a maximal ideal of $D$. The ring of integer-valued polynomials on a subset $E$ of $D$, as well as more general rings of functions from $E$ to $D$, can be viewed as subrings of the product $D^E=\prod_{e\in E}D$. We investigate which maximal ideals of $Int(E,D)$ (or any other subring of $D^E$) come from ultrapowers of the residue class ring $D/M$.