Strongly J-quasi-clean Rings And Strongly GJ-quasi-clean Rings

It is an important way of ring theory to study the structure and properties of rings by the decomposition of elements.A ring R is called a(strongly)clean ring if every element of R can be written as a sum of an idempotent and a unit(that commute).A ring R is called a strongly J-clean ring if every element of R can be written as a sum of an idempotent and an element in its Jacobson radical that commute.An element a in a ring R is called a quasi-idempotent if there exists some k∈UC(R)such that a2=ka.In this paper,we define strongly J-quasi-clean rings and strongly GJ-clean rings by using quasi-idempotents and generalize strongly J-clean rings.The main content of this paper is divided into the following two parts:In the first part,we introduce the concept of strongly J-quasi-clean rings.A ring R is called a strongly J-quasi-clean ring if for any a∈R,there exist a quasi-idempotent q2=kq and j ∈ J(R)such that a = q+j and qj∈jq where k∈UC(R).We study properties of strongly J-quasi-clean rings,the relationship between strongly J-quasi-clean rings and related rings and its extensions.We prove that:(1)Strongly J-clean rings are strongly J-quasi-clean rings,strongly J-quasi-clean rings are strongly quasi-clean rings and their reverses are not true;(2)The ring R is strongly J-clean if and only if R is UJ strongly J-quasi-clean if and only if R is UJ strongly quasi-clean;(3)The ring R is a strongly J-quasi-clean ring if and only if R/J(R)is quasi-Boolean and every quasi-idempotent can lift strongly modulo J(R);(4)The conner rings of strongly J-quasi-clean ring are also strongly J-quasi-clean rings.In the second part,we further give the properties of the generalized Jacobson radical and introduce the concept of strongly GJ-clean rings by using it.A ring R is called a strongly GJ-quasi-clean ring if for any a∈R,there exist a quasi-idempotent q2=kq and b ∈(?)such that a = q+b and qb-bq where k ∈UC(R).We prove that:(1)Strongly GJ-quasi-clean rings are strongly quasi-clean rings;(2)If R is GUJ quasi-clean,then R is GJ-quasi-clean;(3)The ring R is a strongly GJ-quasi-clean and(?)=0 if and only if R is quasi-Boolean if and only if R is strongly regular and strongly GJ-quasi-clean.