Related Ur-clean Properties Of Rings

Ring theory is an important branch of algebra,and clean ring is an important part of ring theory.Since the introduction of clean rings in 1977,they have been studied by many experts and scholars because of their simple structure and many connections with other rings.It has not only enriched the variety of rings,but also deepened the connection between various rings.In this paper,we propose ur-clean rings in non-commutative rings based on the r-clean rings,generalize the properties of clean rings,and study the relationship between ur-clean rings and other important ring classes,and find the r-clean matrices of non-clean decompositions of matrix rings over integers by C++programming.The specific work is as follows.In Chapter 1,the development and significance of clean rings,regular rings,unit regular rings,*-ring,r-clean rings and other clean ring classes are reviewed and the current status of the research on clean rings in the country and abroad are introduced.In Chapter 2,the definition of ur-clean rings and the inscription of elements are presented,the closure of ur-clean rings under direct product and quotient rings and the equivalent proposition of ur-clean rings are proved,and it is investigated whether ur-clean rings can preserve ur-cleanness over polynomial rings,formal power series rings and matrix rings.In Chapter 3,ur-cleanness on*-rings is studied and*ur-clean rings are proposed.Firstly it is shown that*ur-clean elements are not symmetries and examples are given on matrices with 2 degree.Then some equivalent inscriptions of*ur-clean rings are proved with the expansion property of*ur-clean rings.In Chapter 4,the cleanness and r-cleanness of matrix rings M2(Z),the 2 degree matrices over the integer rings are studied,its algebraic characterization is given,and the determination of regular matrix elements and r-clean matrix elements over M2(Z)is implemented by specific algorithms.Then it wae found that the number of r-clean matrices over the[0,10]which is the interval of integers are greater than 13356 by C++programming,which makes many theorems intuitively verified.