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

环是元素的集合，这些元素可以加、减或乘在一起，但不一定相互除以。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。星运算对于研究某些类型的环，特别是它们的乘法性质是一个有用的工具。该项目将星形运算的使用扩展到比目前使用的环更大的类别。这项研究将在浦项科技大学与交换环理论领域的专家康炳均博士合作进行。星运算是满足与主理想相关的某些公理的交换环的理想的闭包运算。通常研究的星形运算是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。星运算也可以用来构造Kronecker函数环，这对于研究赋值环和Zariski-Riemann空间的拓扑是非常有价值的。它们主要是在整环的背景下研究的，对于零因子环的研究相对较少。这个项目将把星运算的使用扩展到具有零因子的交换环，重点是将Kronecker函数环和Zariski-Riemann空间推广到那种设置。这个奖项由东亚和太平洋夏季学院计划资助一名美国研究生的夏季研究，由NSF和韩国国家研究基金会共同资助。

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.