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

Abstract

As applications of non-standard analysis to the study of transcendence in the ring $\mathcal{A}$ of integers modulo infinitely large primes, we extend transcendence criteria by Anzawa--Funakura and Matsusaka--Seki. Using the extension of the criterion by Anzawa--Funakura, we observe conditional relation between asymptotic behaviour of prime gaps and transcendence in $\mathcal{A}$. We also summarise known algebraic and model theoretic results on $\mathcal{A}$ for number theorists, and share topics in transcendental number theory with algebraists and model theorists.