Multiplicative Closures And Lifting Of Relative Regular Elements

Idempotents are the basic research objects in ring theory,and their multiplicative closures and lifting are paid much attention in ring theory.The definition of regular elements is an important generalization of that of idempotents.Regular elements are related to the concepts of regularity of rings,strong π-regularity and generalized inverses of rings.This paper is focus on the study of the multiplicative closures and lifting of relative regulars in ring R.The multiplicative closures of relative regular elements are studied in Chapter 2.Firstly,it is proved that the multiplication of any two tripotent elements in a ring is also tripotent if and only if any two tripotent elements are commutative in a ring.The necessary and sufficient conditions for the multiplication closures of n-potent elements in a ring are obtained.Next,it is shown that the multiplication of any two regular elements in a ring is also regular if and only if the ring is SSP.By this result,the upper triangular matrix ring of order n(n≥2)is not an SSP ring.Then,the multiplication of any two projection elements in a ring is also a projection if and only if any two projection elements are commutative in a ring.In addition,*-SSP rings are introduced,and the equivalent conditions for a ring to be*-SSP are given.It is proved that the multiplication of any two {1,3}-inverse elements in R is also {1,3}-invertible if and only if the multiplication of any two{1,4}-inverse elements in R is also {1,4}-invertible.Then R satisfies the multiplication of any two Moore-Penrose invertible elements is also Moore-Penrose invertible,which implies that R is*-SSP.Finally,the equivalent conditions of multiplication closures of Moore-Penrose inverses are given.The lifting of relative regular elements are studied in Chapter 3.It is shown that if I is a nil ideal of a ring R,x,y∈R,x-xyx∈I and xy=yx,then there exists a∈R{1} such that x-a E I.It is shown that the lifting property of regular elements is not Morita invariant.The lifting of relative regular elements of the trivial extension rings is also explored.Suppose RMR is a bimodule,I is an ideal of R,T=I∝ N is the special ideal of ring S=R∝M,then idempotents in S/T can be lifted to S if and only if idempotents in R/I can be lifted to R.Similar results for regular elements,units,unit-regular elements and clean elements are also obtained.If R is a*-ring,I is closed under the operation*,it is proved that {1,3}-inverses in R/I can be lifted to R if and only if {1,4}-inverses in R/I can be lifted to R.Then projection elements in R/I can be lifted to R.