具有稳定秩1的置换环  

Exchange Rings Having Stable Range One

作　　者：陈焕艮[1] 

机构地区：[1]湖南师范大学数学与计箅机科学学院,长沙410081

出　　处：《数学年刊（A辑）》2008年第1期83-90,共8页Chinese Annals of Mathematics

摘　　要：称环R具有稳定秩1,如果对任意的a,b∈R,aR+bR=R,则存在y∈R,使得a+by∈U(R).证明了置换环有稳定秩1当且仅当对任意的幂等元e∈R,如果aR+b(eR)=R,则存在u,u∈R,使得au+b(ev)=0且(eR)u+(eR)(ev)=eR当且仅当对任意的幂等元e∈R,如果aR+b(eR)=R,则存在u,v∈R,使得as+b(et)=0当且仅当存在z∈eR,使s=uz,t=uz,从而给出这类置换环新的元素刻画.进一步地,证明了如果R是稳定秩1的置换环,对任意的正则元a∈R,2a总可以表示成两个单位的和.最后对具有降链本原分式的置换环R,证明了对任意的a∈R,2a总可以表示成两个单位的和.A ring R has stable range one provided that aR ＋ bR = R with a, b ∈ R implies that there exists a y ∈ R such that a ＋ by ∈U（R）. This paper proves an exchange ring R has stable range one if and only if for any idempotent e ∈ R, aR ＋ b（eR） = R implies that there exist u,v ∈ R such that au ＋ b（ev） = 0 and （eR）u ＋ （eR）（ev） = eR if and only if for any idempotent e ∈ R, aR ＋ b（eR） = R implies that there exist u, v ∈ R such that as ＋ b（et） = 0 if s = uz and t = vz for a z ∈ eR. These give new element-wise characterizations of such exchange rings. Furthermore, the author proves that for any regular a ∈ R, 2a is the sum of two units, where R is an exchange ring having stable range one. Finally, it is proved that for any a ∈ R, 2a is the sum of two units, where R is an exchange ring with artinian primitive factors.

关 键 词：置换环 稳定秩1 单位 

分 类 号：O153.3[理学—数学]