关于循环环的零因子、零化子以及单位群的结构  

On the Structure of Nil-factor、Annihilator and Unit-group of Cyclical Ring

作　　者：张隆辉[1] 石化国[1] ZHANG Long-hui;SHI Hua-guo(Sichuan Vocational and Technical College,Suining 629000,China)

机构地区：[1]四川职业技术学院

出　　处：《数学的实践与认识》2018年第24期288-292,共5页Mathematics in Practice and Theory

基　　金：四川省教育厅自然科学重点项目(11ZA263,18ZA0434)

摘　　要：研究了循环环的零因子、零化子以及单位群的结构,得到的主要结论有:1)若R为无限循环非零乘环,则有R0=φ,Z(R)=0;又设R=,a2=ka(k∈Z,k≠0),若|k|=1,则R*={a,-a};若|k|> 1,则R*=φ.2)设n(> 1)阶循环环R=,a2=ka(k∈Z,0 <k <n), i)如果(k,n)≠1,则有R0=R-{0}, Z(R)=<n/(k,n)a>,|Z(R)|=(k,n),R*=φ; ii)如果(k,n)=1,则有R0={sa|0<s<n,(s,n)≠1},Z(R)=0, R*={sa|0 <s <n,(s,n)=1},|R*|=φ(n);并且R*是循环群的充要条件是:(k,n)=1,且n等于2,4,pα或2pα(p是奇质数).最后,给出了上述主要结论的一个应用.In this paper,authors investgate the structure of nilfactor、annihilator and unit-group of cyclical ring.Main Findings Are:1) If R is an infinite cyclical non-zeromultiplication ring then R0= φ,Z(R) =0;Set R =(a),a2 = ka(k ∈ Z,k≠ 0),if |k| = 1,then R* = {a,-a};if |k|> 1,then R*=φ.2)Let finite cyclicalring R =(a),a2=ka(k∈Z,0 < k < n) of order n(> 1),i)if(k,n) ≠ 1,then R0 = R-{0},Z(R) =<n/(k,n)a>,|Z(R)| =(k,n),R*=φ;ii) if(k,n) = 1,then R0 = {sa|0 < s < n,(s,n) ≠ 1},Z(R)= 0,R* = {sa|0 < s < n,(s,n) = 1},|R*| = φ(n);Further more R* is a cyclical group if and only if:(k,n) = 1,and n equal to 2,4,pα or 2 pα(where p is an odd prime number).Finaly,an application of above maun results is given.

关 键 词：循环环 零因子 零化子 单位群 

分 类 号：O153.3[理学—数学]