The Cleanness Of The Direct Sum Of A Finite Copies Of A Module M

Let R be an associa.tive ring with unity.An element a ∈ R is said to be clean if a = e+u,where e is an idempotent and u is a unit in R.If every element of R is clean then R is called a clean ring.As an important class of rings,the idea of clean rings is derived from the problem of modules elimination.In 1977,clean rings were introduced by Nicholson.Nicholson proved that every clean ring is an exchange ring,and a ring with central idempotents is clean iff it is an exchange ring.In 1994,Camillo and Yu found an important counter example:an exchange ring which is not clean,and proved semiperfect rings and unit-regular rings are clean rings.Semiperfect rings,clean rings and exchange rings are equivalent,when a ring R containing no infinite set of orthogonal idempotents.It is well known that a continuous module is quasi-continuous,and that a quasi-injective module is continuous.Mohamed and Muller proved that continuous modules satisfy the exchange property defined by Crawley and Jonsson.Warfield proved that a module M have exchange property if and only if the endomorphism ring of a module is an exchange ring.In 1994,Mohamed and Muller proved that the endomorphism ring of a,continuous module is an exchange ring.In 2006,Nicholson proved that every continuous module M is clean.If a module M is ∑-CS(in the sense that the direct sum of any number of copies of M is CS),then M is clean.For the E-CS,we find the module that the direct.sum of any number of copies of M is clean.In this paper,the cleanness of the direct sum of a finite copies of a clean module will be given.This paper is composed of four parts.The first part is introduction.The second and third part are the main body of this paper.And the last part is the concluding remarks.In Chapter 1,some of the research background,research significance and main results of this paper are introduced.Such as clean rings and clean modules etc.In Chapter 2,we will give a discussion of the direct sum of a finite copies of a clean module.It is well known that the finite sum of clean modules is clean.On the contrary,such as modules of finite indecomposable decomposit ion,quasi-continuous modules.quasi-discrete modules etc.We can prove that these modules are clean if the direct sum of these modules is clean.The main results are Theorem 2.5,2.10,2.19.Theorem 2.5 If the module M has finite indecomposable decomposition,then the module M is n-∑-clean(or M(n)= M(?)...(?)M is clean)if and only if the module M is clean.Theorem 2.10 If the module M is quasi-continuous,then the module M(n)=M(?)...(?)M is clean if and only if the module M is clean.Theorem 2.18 If the module M is quasi-discrete,then the module M(n)= M(?)...(?)M is clean if and only if the module M is clean.In Chapter 3,the cleanness of some modules of satisfy(D1)will be discussed.We find some modules of satisfy(D1)when we discussing the cleanness of the finitely sum of clean modules.The main results are Proposition 3.3,3.4,3.5.Proposition 3.3 The Rickart module with condition(D1)is clean.Proposition 3.4 The endoregular module with condition(D1)is clean.Proposition 3.5 The quasi-projective module with condition(D1)is clean.The last part is the concluding remarks.The main work of this paper will be summa-rized,and some problems will be put forward.