EAPSI: Extending the Scope of Mathematical Operations used as Tools to Study Rings

环是一组元素的集合，这些元素可以相加、相减或相乘，但不一定要彼此相除。环出现在数学和物理的广泛领域。整数、有理数和多项式都是环的常用例子。星型运算是研究某些类型环的有用工具，特别是研究它们的乘法性质。这个项目将把星操作的使用范围扩大到比目前使用的更大种类的环。此次研究将在浦项工业大学与交换环理论专家姜炳均博士共同进行。星型运算是交换环上满足与主理想有关的某些公理的理想的闭包运算。通常研究的星型运算是b、t和v运算，它们是乘法理想理论中一个活跃的研究领域。星形运算也可以用来构造Kronecker函数环，这对于研究估值上环和Zariski-Riemann空间的拓扑是非常宝贵的。它们主要是在积分域的背景下进行研究的，对于零因子环的研究相对较少。本课题将星形运算的应用扩展到零因子交换环，重点将Kronecker函数环和Zariski-Riemann空间推广到该集合。该奖项是由美国国家科学基金会（NSF）和韩国国立科学研究财团共同资助的东亚太平洋暑期研究所项目，旨在支持美国研究生的暑期研究。

A ring is a collection of elements that can be added, subtracted, or multiplied together, but not necessarily divided by one another. Rings appear in a wide range of areas within math and physics. The integers, the rational numbers, and polynomials are all commonly used examples of rings. A star operation is a useful tool for studying certain types of rings, particularly the properties of their multiplication. This project will extend the use of star operations to a larger category of rings than they are currently used for. This research will be conducted at Pohang University of Science and Technology in collaboration with Dr. Byung Gyun Kang, an expert in the field of commutative ring theory.Star operations are a closure operation on the ideals of a commutative ring satisfying certain axioms related to principal ideals. Commonly studied star operations are the b, t, and v operations, and they are an active area of research in multiplicative ideal theory. Star operations can also be used to construct Kronecker function rings, which are invaluable for studying valuation overrings and the topology of Zariski-Riemann spaces. They are mainly studied in the context of integral domains, and comparatively little work has been done in the case of rings with zero divisors. This project will expand the use of star operations to commutative rings with zero divisors, with an emphasis on generalizing Kronecker function rings and Zariski-Riemann spaces to that setting.This award under the East Asia and Pacific Summer Institutes program supports summer research by a U.S. graduate student and is jointly funded by NSF and the National Research Foundation of Korea.