标题： 马西亚斯拓扑在整环上 显示英文标题

标题： The Macias topology on integral domains

摘要： 在本文中，研究了一种最近在正整数上由集合$\{\sigma_n:n\in\mathbb{N}\}$生成的拓扑，其中$\sigma_n:=\{m: \gcd(n,m)=1\}$在整环上被推广。研究了该拓扑的一些拓扑性质。还探讨了在无限主理想域（不是域）上的该拓扑的性质，并得到了一种新的拓扑证明，证明素元素的无限性（假设单位集是有限的或不是开集），不同于H. Furstenberg风格的证明。最后，提出了一些问题。 显示英文摘要

摘要： In this manuscript a recent topology on the positive integers generated by the collection of $\{\sigma_n:n\in\mathbb{N}\}$ where $\sigma_n:=\{m: \gcd(n,m)=1\}$ is generalized over integral domains. Some of its topological properties are studied. Properties of this topology on infinite principal ideal domains that are not fields are also explored, and a new topological proof of the infinitude of prime elements is obtained (assuming the set of units is finite or not open), different from those presented in the style of H. Furstenberg. Finally, some problems are proposed.