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

We summarise known algebraic and model theoretic results on the ring $\mathcal{A}$ 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.