Mathematical Sciences: "Topics in Commutative Algebra"

本课题主要研究交换代数中的下列问题：(1)整数多项式环上的素理想谱和可逆理想群的结构；(2)多项式环中齐次完全理想的结构kıX，Y，Z！以及与投影平面上的线性曲线系统的联系；(3)可数1维局部域R上的射影线与多项式环之间的积分域的谱结构RıX！以及它的分数域。交换环是具有交换加法和交换乘法的代数结构。这些结构贯穿于数学和代数。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。本课题涉及交换环结构的各种问题。

This project is concerned with the following topics in commutative algebra: (1) the structure of the prime ideal spectra and the groups of invertible ideals in certain rings of integer-valued polynomials; (2) the structure of the homogeneous complete ideals in the polynomial ring kıX,Y,Z! and the connections with linear systems of curves in the projective plane; and (3) the projective line over a countable 1- dimensional local domain R and the structure of the spectra of integral domains between the polynomial ring RıX! and its field of fractions. Commutative rings are algebraic structures possessing a commutative addition and a commutative multiplication. These structures occur throughout mathematics and algebra. Common examples include polynomial rings, and rings of algebraic integers in extension fields of the rationals. This project is conerned with a variety of questions on the structure of commutative rings.