Mathematical Sciences: Topics in Commutative Ring Theory

本研究涉及交换结合环和代数领域中的以下几个问题。第一个是二维正规局部区域上的二元正规点和特异素因子。这里的一个具体问题是，满足Muhly条件(N)的二维正规局部域是否必然有扭因子类群。第二个主题涉及由多项式环扩张收缩的理想的有限生成。一个有趣的问题是，一维相干域是否具有其积分闭包是Prufer域的性质。第三个主题涉及环和域扩张，它们不是有限生成的，但具有每个真子扩张是有限生成的性质。交换环理论中的其他主题涉及到Picard群在环扩张下的行为，控制论中感兴趣的环的结构，以及二维正则局部域中相邻理想的重数。交换环是具有交换加法和交换乘法的代数结构。这些结构在整个数学和代数中都存在，常见的例子包括多项式环，以及有理数的扩域中的代数整数环。这个项目涉及到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.