一类特殊的有限循环环  

A Special Sort of Finite Cyclic Rings

作　　者：张隆辉[1] 

机构地区：[1]四川职业技术学院学报编辑部,四川遂宁629000

出　　处：《数学的实践与认识》2013年第6期235-239,共5页Mathematics in Practice and Theory

基　　金：四川职业技术学院科研项目(2009205);四川省教育厅自然科学重点项目(11ZA263)

摘　　要：证明了一类n阶(n=P_1P_2…p_m,p_i(i=1,2,…,m)互异为素数)环是有限循环环,并讨论了他们的结构及相关性质,最后给出了这类n阶环有零因子或有子域的充要条件.主要结果:P_1P_2…P_m阶环共有2~m个,它们是(p_(1~k_1) p_(2~k_2)…p_(m^k_m)Z)/(p_(1~k_1+1)p_(2~k_2+1)…p_(m^k_m+1)Z),其中k_i=0或1,1≤i≤m;阶是n=P_1P_2…p_m的环R可唯一分解为m个素数阶理想的直和,即R=〈α〉=(?);含pi(1≤i≤m)阶子域的P_1P_2…P_m阶环共有2^(m-1)个,它们是p_(1~k_1) p_(2~k_2)…p_(m^k_m)Z)/(p_(1~k_1+1)p_(2~k_2+1)…p_(m^k_m+1)Z),其.中k_i=0,k_j=0或1,1≤j≤m,j≠i.In this paper, we prove that a ring with order n （ where n=PiP2…pm,pi（i=1,2,…,m） are distinct primes） is a finite cycle ring. The structure of finite cycle rings and some related properties are discussed. Finally, some sufficient and necessary conditions for a finite cycle rings that have zero divisors or subfields are obtained.The main results： there are a total of 2m rings with order pip2 …Pm ,which are （p1k1p2k2…pmkmZ）/（p1k1＋1p2k2＋1…pmkm＋1Z）; the ring R with order n=pip2 pmPro carl be uniquely decom- posed into the direct sum of m prime order ideals, which is, R=（a）=m∑i=1〈n/pia）; there are a rings with orderp1p2…pmwhich each has a subfield with order pi（1 〈 i 〈 m） , which axe （p1k1p2k2…pmkmZ）/（p1k1＋1p2k2＋1…pmkm＋1Z）,where ki=0,kj=0 or1,1≤j≤m,j≠i.

关 键 词：单素因子环 循环环 直和分解 零乘环 零因子 子域 

分 类 号：O153.3[理学—数学]