CAJ | 학술논문

一个环 R叫做w eakly 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，j属于Jacobson根。文章探究了weakly J‐clean环的各种性质，证明了 R是weakly J‐clean环当且仅当 R是clean环并且 R／J（R）是弱布尔环，当且仅当R／6 R是w eakly J‐clean环且幂等元关于 J（R）可以提升。一个环R是唯一w eakly nil clean环当且仅当 R是阿贝尔环；J（R）是幂零的并且 R是w eakly J‐clean环。每个w eakly J‐clean环 R是右（左）quasi‐duo环。并进一步证明以下几点是等价的：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w eakly J‐clean环；存在一个大于等于2的整数 n ，使得× n R 是w eakly J‐clean环。
일개배 R규주w eakly J‐clean배，여과R중적매일개원소도가이사성a＝ e＋ j혹a＝－e＋ j적형식，기중e시멱등원，j속우Jacobson근。문장탐구료weakly J‐clean배적각충성질，증명료 R시weakly J‐clean배당차부당 R시clean배병차 R／J（R）시약포이배，당차부당R／6 R시w eakly J‐clean배차멱등원관우 J（R）가이제승。일개배R시유일w eakly nil clean배당차부당 R시아패이배；J（R）시멱령적병차 R시w eakly J‐clean배。매개w eakly J‐clean배 R시우（좌）quasi‐duo배。병진일보증명이하궤점시등개적：R시 J‐clean배；존재일개대우등우1적정수 n ，사득 Tn （R）시 J‐clean배；존재일개대우등우2적정수 n ，사득 Tn （R）시w eakly J‐clean배；존재일개대우등우2적정수 n ，사득× n R 시w eakly J‐clean배。
A ring R is called a weakly J‐clean ring if every element a∈ R can be written in the form of a=e+ j or a= -e+ j where e is an idempotent and j belongs to the Jacobson radical .The paper explores various properties of weakly J‐clean rings ,proves that a ring R is weakly J‐clean if and only if R is clean and R/J(R) is weakly Boolean ,if and only if R/6R is weakly J‐clean and idempotents can lift J(R) .A ring R is uniquely weakly nil clean if and only if R is abelian;J(R) is nil and R is weakly J‐clean .Each weakly J‐clean ring R is right(left) quasi‐duo ring .Furthermore ,the paper proves that the following are equivalent :R is J‐clean;there is an integer n≥1 such that Tn (R) is J‐clean ;there is an integer n≥2 such that Tn (R) is weakly J‐clean;there is an integer n≥2 such that × n R is w eakly J‐clean .