PIMD上一元多项式环的素理想分类

Classification of the Prime Ideals of Polynomial Ring Over PIMD

设R是主理想整环,若R有无穷多个极大理想,则称R是Principal Ideal Maximal Domain,简称为PIMD.设x是PIMD上的未定元,R[x]是R上的一元多项式环.依据整环的基本理论和唯一分解环的结构理论,研究R[x]的素理想和极大理想,推证了R[x]的任一主理想都不是极大理想,给出了构造R[x]的极大理想的一种方法,得到了R[x]的素理想是极大理想的条件,最终给出R[x]的素理想分类定理.

Let R be a principal ideal domain. If R has infinite maximal ideals, then R is called a Principal Ideal Maximal Domain, for short, PIMD. Let x be an indeterminate over the PIMD R and R[x] denote the polynomial ring over R. In this thesis, according to the basic theory of domain and the structural theory of the unique factorization domain, we study the prime ideals and maximal ideals of R[x]. We deduce that none of the principal ideal is maximal in R[x], and put forward a way to construct the maximal ideals of R[x], determine the conditions of prime ideals of R[x] as the maximal ideals, and ultimately confirme the classification of the prime ideals and maximal ideals of R[x].