Weakly JU Rings?Feckly-reduced Rings And PR-rings

The related theory of Jacobson of rings is a good extension of Artin ring,which is of great significance to our study of ring structure.At the same time,prime also plays an important role in our study of ring structure theory.This article mainly studies the basic properties of weakly JU rings,feckly-reduced rings and PR-rings and some of their ring extensions.This article mainly consists of the following parts:Chapter 1:We introduce the historical backgrounds,developments and research status of weakly JU rings,feckly-reduced rings and PR-rings,and briefly explain the main work and important results of this article.Chapter 2:We mainly introduced the concepts of ring,such as JU rings,UU rings,weakly UU rings,weakly JU rings,reduced rings,symmetric rings,reversible rings,semi-commutative rings,feckly-reduced rings,feckly-reversible rings,feckly-semicommutative rings,expansion and some results which were frequently used in the sequel.Chapter 3:This chapter mainly studies some characterizations of weakly JU rings and investigates their extension properties.The main results are the following:(1)R is a JU ring if and only if R is a weakly JU ring and 2 belongs to J(R);(2)Let I be the nil ideal of ring R,then R/I is a weakly JU ring if and only if R is a weakly JU ring;(3)Let R be a ring,then T(R,M)is a weakly JU ring if and only if R is a weakly JU ring.Chapter 4:This chapter mainly studies the equivalent characterization of feckly-reduced rings,and discusses the relationship between feckly-reduced rings and related rings.We mainly prove that the concepts of feckly-reduced rings,feckly-reversible rings,feckly-symmetric rings and feckly-semicommutative rings are equivalent.Chapter 5:This chapter first mainly puts forward the concept of PR-ring.A ring R is called a PR-ring if every elem chapter ent a?R can be written in the form of a=r+p,where r is a regular element and p belongs to the prime radical.This chapter discusses some properties of PR-rings.Chapter 6:We give a summary about weakly JU ring,feckly-reduced ring and PR-ring in this article,and make further prospects for the future research directions of these types of rings.