Some Researches On Ore Extension Rings And Their Properties

In recent years, whether the properties of rings can be preserved in their polyno-mial extension rings has become an important topic in the ring theory. As a branch of the subject, the research on the properties of Ore extensions have become one of the hot topics in algebra research. On the basis of existing research, this paper will study the weak symmetric, weak zip, nilpotent p.p. and nilpotent Baer property of the Ore extension R[x;α,δ] by weakening conditions of some conclusions. Since generalized semicommutativity and generalized symmetry of the skew polynomial rings have been researched, we will continue to investigate these properties of the differential polynomial rings and Ore extensions. This article mainly consists of the following components:Chapter 1:We introduce the background and development process of the property research of Ore extensions, and briefly generalise some research contents and results in the paper;Chapter 2:We mainly introduce the definitions of some rings and the related conclusions;Chapter 3:We investigate the weak symmetric, weak zip, nilpotent p.p. and nilpotent Baer property of the Ore extensions under the condition of (α,δ)-weakly rigid rings, respectively. By using the itemized analysis method on polynomials, we prove that if R is (α,δ)-weakly rigid and semicommutative, then R[x;α,δ] is weak symmetric (resp., weak zip, nilpotent p.p., nilpotent Baer) if and only if it is weak symmetric (resp., weak zip, nilpotent p.p., nilpotent Baer). These results unify and extend nontrivially the previously known results;Chapter 4:In this chapter, we study the nilpotent p.p., nilpotent Baer and weak McCoy property of the Ore extensions under the condition of (α,δ)-condition rings;Chapter 5:We investigate the generalized semicommutativity and generalized symmetry of the differential polynomial rings and some properties of polynomial rings. Under certain conditions, we present the necessary and sufficient conditions for the differential polynomial ring R[x;δ] is a nil-semicommutative (resp., weakly semicom-mutative, GWS) ring;Chapter 6:Given a summary about the conclusions in this article, we also fore- casted some related theoretical research on the property of Ore extensions in the future.