Notes on Algebraic Properties and Non-Standard Analysis of the Ring of Integers Modulo Infinitely Large Primes

TL;DR

This paper reviews algebraic and model theoretic properties of the ring of integers modulo infinitely large primes, extending transcendence criteria and applying non-standard analysis to transcendence studies.

Contribution

It provides a summary of known results and extends transcendence criteria to facilitate the application of non-standard analysis in transcendence theory.

Findings

Extended transcendence criteria by Anzawa--Funakura and Matsusaka--Seki.

Applied non-standard analysis to transcendence problems.

Summarized algebraic and model theoretic results on the ring of integers modulo large primes.

Abstract

We summarise known algebraic and model theoretic results on the ring $\mathscr{A}$ of integers modulo infinitely large primes for number theorists, and share topics in transcendental number theory with algebraists and model theorists. In particular, we extend transcendence criteria by Anzawa--Funakura and Matsusaka--Seki in order to show application of non-standard analysis to the study of transcendence.