Mathematical Sciences: Topics in Commutative Ring Theory

本研究涉及在交换结合环和代数领域的以下主题的工作。首先是二维法向局部域上的两分法向点和例外质因数。这里的一个具体问题是，满足Muhly条件(N)的二维法向局部域是否必然存在扭转因子类群。第二个主题是关于由多项式环扩展收缩的理想的有限生成。这个领域中一个有趣的问题是，一维相干域是否具有其积分闭包为普吕弗域的性质。第三个主题涉及不是有限生成的环和字段扩展，但具有有限生成每个适当子扩展的属性。交换环理论中的其他主题涉及环扩展下Picard群的行为，控制论中感兴趣的环的结构，以及2维正则局部区域内邻近理想的多重性。交换环是具有交换加法和交换乘法的代数结构。这些结构出现在整个数学和代数中，常见的例子包括多项式环和有理数扩展域中的代数整数环。这个项目是关于交换环结构的各种问题。

This research concerns work on the following topics in the area of commutative associative rings and algebras. The first is birational normal spots and exceptional prime divisors over a 2- dimensional normal local domain. A specific question here is whether a 2-dimensional normal local domain that satisfies Muhly's condition (N) necessarily has torsion divisor class group. The second topic concerns finite generation of ideals contracted from a polynomial ring extension. An interesting question in this area is whether a 1-dimensional coherent domain has the property that its integral closure is a Prufer domain. The third topic concerns ring and field extensions that are not finitely generated but have the property that each proper subextension is finitely generated. Other topics in commutative ring theory concerned with the behavior of the Picard group under ring extension, the structure of rings of interest in control theory, and the multiplicity of adjacent ideals in a 2- dimensional regular local domain will be considered. Commutative rings are algebraic structures possessing a commutative addition and a commutative multiplication. These structures occur throughout mathematics and algebra and common examples include polynomial rings, and rings of algebraic integers in extension fields of the rationals. This project is concerned with a variety of questions on the structure of commutative rings.