Some Rings That Similar To VNL Rings

In recent years, local rings and VNL rings have become an important research object in algebra. It mainly contains the following research directions. Firstly, one can study the properties of t.he local rings and VNL rings directly, characterize the radical properties of these rings; Secondly, it is studied for some problems of extensions about these two rings; Thirdly.we can also put forward the definition of the new rings which similar to the definition of local rings and VNL rings and study these new rings’ properties. We research on the third direction.We put forward the definition of almost strongly regular rings. almost weakly regular rings,almost Π-regular rings. almost strongly Π-regular rings. and discuss the relationship between these rings, study their properties and extensions. This thesis mainly consists of the following components:Chapter 1: We introduce the background, development process and research status of the local rings. VNL rings and some related rings, briefly put forward the process of the question in this paper. and briefly summarize some important work and results in the literature.Chapter 2: We introduce some essential concepts, such as strongly regular rings, weakly regular rings. unit-regular rings, Π-regular rings. strongly Π-regular rings. almost unit-regular rings. SVNL rings, and some results which were frequently used in the sequel.Chapter 3: We bring in the concept of almost strongly regular rings. A ring is called almost strongly regular ring, if for any a ∈R. a or a is strongly regular. Almost strongly regular rings is between local rings and VNL rings. The main result are the following: (1) Every reduced VNL ring is an almost strongly regular ring; (2) Every abelian VNL ring is an almost strongly regular ring; (3)Π{Rα : α∈I } is an almost strongly regular ring if and only if there existα0 ∈I such that Rα0 is an almost strongly regular ring and for all α∈I - α0,Rα is strongly regular ring: (4) The direct limit of almost strongly regular ring is also an almost strongly regular ring.Chapter 4: In this chapter, we bring in the concept of almost weakly regular rings and con-summate the relationship between local rings. VNL rings, almost strongly regular rings. almost weakly regular rings. almost Π-regular rings, almost strongly Π-regular rings. The relationship is showed in the following diagram. We also study extensions’ problems about these rings.(?)Chapter 5: Given a summary about related rings we studied which similar to VNL rings in this article, we roughly forecasted their possible applications and other research in the future.