关于Weakly J~#-clean环

On Weakly J~#-clean Rings

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 aER can be written as a=e＋j or a=-e＋j, in which e is idempotent and j^n 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/1 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 T, （R） is a J^# -clean ring, there is an integer n≥2 such that T,,（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.