关于Weakly Almost Clean环（英文）  

Weakly Almost Clean Rings

作　　者：孟文静[1] 陈焕艮[1] 

机构地区：[1]杭州师范大学理学院,浙江杭州310036

出　　处：《杭州师范大学学报（自然科学版）》2015年第4期418-422,共5页Journal of Hangzhou Normal University(Natural Science Edition)

基　　金：Supported by the Natural Science Foundation of Zhejiang Province(LY13A010019)

摘　　要：定义了weakly almost clean环.交换环R叫做weakly almost clean环,如果对于任意一个元素x∈R可以写成x=r+e或x=r-e的形式,其中r∈reg(R)且e∈Id(R).首先,对于环Ri的非空集合{Ri},证明了直和R=∏i∈IRi为weakly almost clean当且仅当存在m∈I使Rm为weakly almost clean且对所有的n≠m,Rn为almost clean.然后,设R是一个环且M为一个R-模,得到了R和M的平凡扩张R(M)为weakly almost clean当且仅当每个x∈R可以写成x=r+e或x=r-e的形式,其中r∈R-(Z(R)∪Z(M))且e∈Id(R).进而推广了almost clean环的相应结果.This paper defines weakly almost clean rings.A commutative ring Ris a weakly almost clean ring if every element x∈Rcan be written in the formx=r＋e or x=r-e where r∈reg（R）and e∈Id（R）.Firstly,for a nonempty collection{Ri}of rings Ri,the product R =∏if and onl∈IRiis weakly almost clean iy if there exists m∈Isuch that Rmis weakly almost clean and Rnis almost clean for all n≠m.Further,let Rbe a ring and M be an R-module,the trivial extension R（M）of Rand Mis weakly almost clean if and only if each x∈Rcan be written in the formx=r＋e or x=r-e where r∈R-（Z（R）∪Z（M））and e∈Id（R）.These extend the corresponding results on almost clean rings.

关 键 词：CLEAN环 零因子 正则元 幂等元 ALMOST CLEAN环 

分 类 号：O153.3[理学—数学]