摘要
文章给出了n-幂等元的定义,它是幂等元的一个推广,讨论n-幂等元的若干性质,证明了当e是环R的n-幂等元时,e^(n-1)和1-e^(n-1)是环R的幂等元,并且对任意x∈R,t=e+(1-e^(n-1))xe^(n-1)是环R的n-幂等元.给出n-幂等元的一个应用,得出当e∈R是n-幂等元时,环R有左理想分解RR=Re^(n-1)⊕R(1-e^(n-1))和右理想分解RR=e^(n-1)R⊕(1-e^(n-1))R.最后,研究等式xR=yR成立的充要条件,其中x是环R的n-幂等元且y是环R的m-幂等元.
The definition of n-idempotent is given,which is a generalization of idempotent.If e∈R and en=e(n≥2),then e is said to be an n-idempotent of R.Some properties of n-idempotents are discussed,it is proved that e^(n-1) and 1-e^(n-1) are idempotents of R,and t=e+(1-e^(n-1))x e^(n-1) is n-idempotent of R for any x∈R,when e is an n-idempotent of R.An application of n-idempotent is given,R has a left ideal decomposition RR=Re^(n-1)⊕R(1-e^(n-1))and a right ideal decomposition RR=e^(n-1)R⊕(1-e^(n-1))R,when e∈R is an n-idempotent of R.Finally,sufficient and necessary conditions for the equation xR=yR holds are investigated with x n-idempotent and y m-idempotent.
作者
何东林
HE Donglin(School of Mathematics and Information,Longnan Normal University,Chengxian Gansu 742500)
出处
《甘肃高师学报》
2024年第5期7-11,共5页
Journal of Gansu Normal Colleges
基金
甘肃省高等学校创新基金项目“Rickart环和模及相关问题的研究”(2021B-364).
关键词
n-幂等元
子环
幂等元
左理想分解
右理想分解
n-idempotent
sub ring
idempotent
left ideal decomposition
right ideal decomposition
