The Unit Groups And Cubic Mapping Graphs Of Quotient Rings Of The Imaginary Quadratic Rings

Let K = Q((?)) , where Q is the rational number field and d is a square-free integer other than 0 and 1. Then K is called a quadratic field and it has degree 2 over Q. We use Rd to stand for the ring of algebraic integers of K. If d <0, then K is an imaginary quadratic field, while Rd is called an imaginary quadratic ring.From the work of H. M. Stark in 1967, we know that there are only finite negative integers d such that the imaginary quadratic ring Rd is a unique-factorization domain, namely d = -1,-2,-3,-7,-11,-19,-43,-67,-163 . For an arbitrary prime element (?)?Rd , and a positive integer n , the unit groups of Rd/<(?)n> were determined for the case d?-1 by J. T. Cross in 1983, Gaohua Tang and Huadong Su in 2010, and the case d =-2 by Yangjiang Wei in 2016, respectively. Furthermore,the structure of cubic mapping graphs of the quotient rings of Rd for the case d = -1 was investigated by Yangjiang Wei in 2016. In this thesis,we investigate the unit groups and the cubic mapping graphs of the quotient rings of Rd for d = -3,-7,-11,-19,-43,-67,-163.In Chapter 1, we introduce the background, some basic definitions and propositions.In Chapter 2, we study the unit groups of Rd /<(?)n>, where (?) is a prime in Rd,and n is an arbitrary positive integer.In Chapter 3, we research the structure of cubic mapping graphs of Rd/<?>,where y is a non-invertible element in Rd .We obtain the number of fixed points,the indegree of the vertices 0?1.