R-Clean Morphisms And R*-Clean Morphisms

The clean ring was first proposed by the American mathematician Nicholson,which originated in the study of the problem of the eliminate of modules.But since it was born,it has shown great vitality and has gradually become a hot topic in the field of ring theory.In the study of the properties of clean rings,we found that there is a close relationship between clean rings and regular rings.Then in 2011 Nahid and Ebrahim proposed the concept of r-clean ring.In this paper,on the basis of these studies,we continue to explore the properties of r-clean ring,and put forward the definition of r-clean morphism and r*-clean morphism.Some results and corresponding properties are also given.This paper has three parts.In the first part,we put forward the background and the development of this paper.We introduce the structure,the research ideas and the main conclusions of the paper.The second part is the main part of this paper,a special kind of morphisms,r-clean,morphisms.Let R is a associative ring with unit.A element a in the HomR[M,N]is called a r-clean morphism if there exist a idempotent morphism ? and a regular morphism r in HomR[M,N]imply that ? = ? + r.We have example show that the r-clean morphisms are the true extension of clean morphisms.After giving some properties and basic theorems of r-clean morphisms,we discuss the influence of homomorphism on the r-clean proper-ties of morphism groups.It.is proved that r-clean morphism multiplicate the left or right invertible morphism is also the r-clean morphism.Using the natural matrix representation of morphism set[M,N],discuss the conditions of the morphism between direct sunmand is r-clean morphism.We obtain the necessary and sufficient condition for the morphism of diagonal form to be r-clean.Finally,by combining the properties of direct N-injective modules and direct M-projective moduled for the morphism set[M,N],we prove that,when M is direct N-injective module,then[M,N]is r-clean;when N is direct M-projective module,then[M,N]is r-clean.The third part,according to the regular morphisms and the semiregular morphisms we define the r*-clean morphisms.A element a in the HomR[M,N]is called a r*-clean mor-phism if there exist a idempotent morphism ? and a regular morphism ? in HomR[M,N]imply that ?=?+?.We proved that r*-clean morphism multiplicate the left or right invertible morphism is also the r*-clean morphism.Similar to the second part of the dis-cussion on M ?M₁???M₂,N=N₁???N₂,we get some results of r*-clean morphisms,and proved that when given left R-module,M,N,and J?[M,N]?= 0,then[M,N]is r-clean if and only if[M,N]is r*-clean.