The Iteration Digraphs And Unit Groups Of Factor Rings Over Complex Quadratic Rings

Let Q be the set of rational number. For a given square-free integer d other than 0 and 1, let K= Q((d^1/2)). Then K is called a quadratic field and it has degree 2 over Q. We denote by Ok the ring of algebraic integers of K. If d< 0, then K is called the complex quadratic field, and Ok is a complex quadratic ring. Specially, the ring Ok is exactly the ring of Gaussian integers Z[i] when d=-1. Rings Ok play important roles in commutative algebra.For a given set S and a mapping f:S→S, we define the iteration diagraph G, whose vertices are all the elements of S and there exists a directed edge from a to β if and only if β= f（α） for α,β ∈ S. In this thesis, we apply the ring theory, number theory and graph theory to study the iteration digraphs of factor rings over complex quadratic rings. Moreover, we completely determine the structure of unit groups of the factor rings over the ring of integers of K= Q(（-7）^1/2). The main contents are as follows.First of all, we summarize the background of our research and review the re-search progresses of the related issues. In addition, we give the contents of each chapter and describe the lemmas which are used in our thesis. In chapter 2, we study the cubic iteration digraphs over Z[i]/<γ>, where γ is the power of an prime of Z[i]. Also, the fifth power iteration digraphs over Z[i]/<n>, where n> 1 is an arbitrary integer, are investigated in chapter 3. In these two chapters, we obtain the the number of fixed points, the in-degree of the elements 0 and 1, and completely characterize the semiregularity of our given iteration digraphs. Moreover, the linear iteration digraphs over Ok/pOk are stated in chapter 4, where p is a rational prime integer. We study the structure of the given linear digraphs in terms of p. Finally, for the purpose of our follow research, we study the unit groups of the factor rings over the rings of integers of K= Q(（-7）^1/2).