On relationships between algebraic properties of groups and rings in some model-theoretic contexts

On the decidability of the theory of modules over the ring of algebraic integers

We prove that the theory of all modules over the ring of algebraic integers is decidable.

Non-definability of rings of integers in most algebraic fields

We show that the set of algebraic extensions $F$ of $\mathbb{Q}$ in which $\mathbb{Z}$ or the ring of integers $\mathcal{O}_F$ are definable is meager in the set of all algebraic extensions.

Rings of Algebraic Numbers in Infinite Extensions of $\Q$ and Elliptic Curves Retaining Their Rank

We show that elliptic curves whose Mordell-Weil groups are finitely generated over some infinite extensions of $\Q$, can be used to show the Diophantine undecidability of the rings of integers and bigger rings contained in some infinite extensions of rational numbers.

环与布尔代数作为代数理论

Rings and Boolean Algebras as Algebraic Theories

本文提出通过仿射和超仿射代数理论统一表示交换环和布尔环，并探讨了其与布尔集合及多维布尔代数的关系。

Boolean metric spaces and Boolean algebraic varieties

The concepts of Boolean metric space and convex combination are used to characterize polynomial maps in a class of commutative Von Neumann regular rings including Boolean rings and p-rings, that we have called CFG-rings. In those rings, the study of the category of algebraic varieties (i.e. sets of solutions to a finite number of polynomial equations with polynomial maps as morphisms) is equivalent to the study of a class of Boolean metric spaces, that we call here CFG-spaces.

关于无限大素数模整数环的代数性质和非标准分析的注记

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

本文探讨无限大素数模整数环的代数与模型论性质，结合非标准分析方法扩展超越数理论中的判别准则。