The Generalizations Of Clean Rings

In 1977,Nicholson proposed clean rings.A lot of algebra scholars at home and abroad begin to research clean rings.And as the generalization of unit regular rings,clean rings have practical application.Following clean rings,Xiao and Zhang generalized clean rings to n-clean rings and generalized clean rings.In section two,firstly the notion of generalized n-clean rings is introduced.These rings are shown to be a natual generalization of n-clean rings and generalized clean rings.A ring R is called generalized n-clean if (?)_x∈R,x =w+u₁+u₂+...+u_n ,where w is a unit regular element and u₁,u₂,...,u_n∈U(R).Clearly n-clean rings and generalized clean rings must be generalized n-clean rings.And an example that generalized n-clean rings need not be generalized clean rings will be given.We will prove that if idempotents are central generalized n-clean rings are (n+1)-clean rings.The basic properties of generalized n-clean rings and the relationship between generalize n-clean rings and n-good rings will be discussed.Secondly,we will discuss matrix extensions of generalized n-clean rings.It is known that any matrix ring is 3-clean.And in [20],it is proved that the row and column-finite matrix ring is 2-clean.Thus any matrix ring is 2-clean.We will decompose the matrix and give another proof.Finally,we will mainly discuss polynomial extensions of generalized n-clean rings.It will be proved that for any 2-primal ring R ,the polynomial ring R[ x] is not generalized n-clean.In section three,semiclean rings will be extended to general rings.And semiclean general rings will be defined.A general ring I is called semiclean if (?)_x∈I,x = a+q,where a is periodic,i.e.,a^k = a^l,a∈I for some positive integers k and l ( k≠l) and q∈Q.In the other hand,some properties of semiclean rings will be dicussed.And it is proved that the matrix ring over semiclean ring is semiclean.