摘要：
一个环R叫做weakly J#-clean环,如果R中的每一个元素都可以写成a=e+j或a=-e+j的形式,其中e是幂等元,jn属于Jacobson根.在这篇文章中我们证明了R是weakly nil-clean环当且仅当R是weakly J#-clean环并且J(R)是幂零的.如果I是幂零的,那么R是weakly J#-clean环当且仅当R/I是weakly J#-clean环.环R是weakly J#-clean环当且仅当R/P(R), R×M和幂级数环R[[x]]分别为weakly J#-clean环.更进一步我们证明以下几点是分别等价的:R是J#-clean环;存在一个大于等于1的整数n,使得Tn(R)是J#-clean环;存在一个大于等于2的整数n,使得Tn(R)是weakly J#-clean环.而且,R是J#-clean环;存在一个大于等于1的整数n,使得×nR是J#-clean环;存在一个大于等于2的整数n,使得×nR是weakly J#-clean环.特殊的,阐述了在某种条件下S=R[D,C]是weakly J#-clean环.%A ring R is called a weakly J#-clean ring if for any a∈R can be written as a=e+j or a=-e+j,in which e is idempotent and jn belongs to Jacobson radical.This article proves a ring R is a weakly nil-clean ring if and only if R is weakly J#-clean ring and J(R) is nilpotent.If I is nilpotent,then R is a weakly J#-clean ring if and only if R/I is a weakly J#-clean ring.A ring R is a weakly J#-clean ring if and only if R/P(R),R×M,power series ring R[[x]] are weakly J#-clean rings respectively.Furthermore,it is proved that the followings are equivalent respectively,R is a J#-clean ring,there is an integer n≥1 such that Tn(R) is a J#-clean ring,there is an integer n≥2 such that Tn(R) is a weakly J#-clean ring.Also,R is a J#-clean ring,there is an integer n≥1 such that ×nR is a J#-clean ring,there is an integer n≥2 such that ×nR is a weakly J#-clean ring.In particular,S=R[D,C] is weakly J#-clean under certain conditions is exposed.
