On Power Series Rings Having McCoy-like Or Armendariz-like Conditions

In recent years, power series rings have become an important research object in algebra. It mainly contains the following research directions. Firstly, for the formal of power series rings, the general properties of known power series rings are directly discussed or extend to a new ring to study its properties and extension. Secondly, the skew power series are constructed based on the power series rings with the homomor-phism ring. Thirdly, the generalized properties and the extending structures of power series rings are researched. This paper is focused on the first direction and discussed the concepts, properties and extensions of rings, such as power series weak McCoy rings, power series π-Armendariz rings, power series J-Armendariz rings. Therefore,this thesis mainly consists of the following parts:Chapter 1. We introduce the background, development process and research status of power series rings then summarize some important work and results in the literature.Chapter 2. Many essential concepts and frequently used theories are introduced such as power series Armendariz rings, weak power series Armendariz rings, nil power series rings, power series McCoy rings, NI rings, α-compatable rings, nil semicommu-tative rings, nilpotent p.p. rings and weakly zip rings and some results which were frequently used in the paper.Chapter 3, The concept of a power series weak McCoy ring is introduced to extend the research of power series McCoy rings. We discuss their basic properties and basic extensions of power series weak McCoy rings.Chapter 4, We introduce the concept of a power series 7r-Armendariz ring. We study the extensions of the power series π-Armendariz rings, which shows that if R is of bounded index, an NI ring and α-compatible, then R[x;α] is π-Armendariz. Meanwhile. we investigate the properties of the nilpotent p.p.property and weak zip property over a power series ring.Chapter 5, We introduce the concept of a power series J-Armendariz ring with the knowledge of Jacobson radical. We extend the research of power series Armen-dariz rings. At the same time, we discuss the relationship between the power series J-Armendariz rings and other rings. We investigate some properties and extensions of the power series J-Armendariz rings.Chapter 6, A summary is given including the power series rings having McCoy-like or Armendariz-like conditions and their possible related research.