On the Ring Theory of Skew Generalized Power Series

摘要

The thesis is mainly about the rings of skew generalized power series，which con-sists of five chapters.
　　In Chapter 1，background and basic concepts are given.
　　Let (S,≤) be a strictly ordered monoid and for some ring R，ω ：S → End(R) a monoid homomorphism. In Chapter 2，we introduce skew generalized power series (S,ω)-Armendariz ring，and study the structure of the set of its nilpotent elements. Moreover，we obtain various necessary and sufficient conditions for a ring to be (S,ω)-nil-Armendariz. Our results unify and generalize the corresponding results of the poly-nomial extension，power series extension of rings and the generalized power series rings.
　　In Chapter 3，some characterizations of three kinds of generalized quasi-Armendariz rings，namely S-quasi-Armendariz，(S,ω)-quasi-Armendariz and linearly (S,ω)-quasi-Armendariz rings，are given. The existence of S-quasi-Armendariz and (S,ω)-quasi-Armendariz rings are discussed. We prove a sufficient condition for a ring to be (S,ω)-quasi-Armendariz. Finally，we show that：(1) If R is a linearly (S,ω)-quasi-Armendariz ring，U is a nonempty subset of R，A = lR(U） is a two-sided ideal of R，and ωs|U is surjective for all s ∈ S，then R/A is a linearly (S,ω)-quasi-Armendariz；(2) For a two-sided ideal I of R，if R/I is a linearly (S,ω)-quasi-Armendariz ring and I is a semiprime ring without identity，then R is linearly (S,ω)-quasi-Armendariz.
　　In Chapter 4，the Malcev-Numann ring is mainly discussed. We prove that the Malcev-Neumann ring R*((G)) is a P S-ring，when R is PS . Also，we prove that the Malcev-Neumann ring R*((G)) is a P F ring，if and only if for any two G-indexed subsets A and B of R such that B ? annR (A)，there exists c ∈ annR (A) such that bc=b for all b∈B. Then it follows that R*((G)) is a P P ring if and only if so is R.
　　In Chapter 5，we introduce and study strongly right S-α-reversible ring. Although the n-by-n upper triangular matrix ring is not strongly right S-α-reversible for any ring R with identity and n ≥ 2，we show that a special subring of upper triangular matrix ring over a reduced ring is a strongly right S-α-reversible under some additional conditions. Finally，some examples of strongly S-α-reversible rings are presented.

目录

封面

目录

英文摘要

List of Symb ols

Chapter 1 Background and preliminaries

§1.1 Some constructions and basic concepts of rings

§1.2 Rings of generalized power series

§1.3 Rings of skew generalized power series

Chapter 2 Armendariz rings of skew generalized power series: Nilpotent elements

§2.1 Introduction

§2.2 (S, ω)-nil-Armendariz rings

§2.3 Weak annihilator of (S,ω)-nil-Armendariz rings

Chapter 3 A generalization of Quasi-Armendariz rings

§3.1 Introduction

§3.2 Generalized power series quasi-Armendariz rings

§3.3 Characterizations generalized power series quasi-Armendariz rings via annihilators

§3.4 (S, ω)-Quasi-Armendariz rings

§3.5 The (S,ω)-quasi-Armendariz condition and ring extensions

§3.6 Linearly (S,ω)-quasi-Armendariz rings

Chapter 4 On some properties of Malcev-Neumann rings

§4.1 Introduction

§4.2 PS-rings

§4.3 APP-rings

§4.4 PF and PP-rings

§4.5 Zip-rings

Chapter 5 Monoid rings over α-reversible rings

§5.1 Introduction

§5.2 Extensions of reversible rings

§5.3 Monoid rings over α-reversible rings

参考文献

Publications

致谢